Additional Topics in Analytic Geometry

Transcription

1 bar1969_ch11_ qd 17/1/08 11:43 PM Page 961 Pinnacle ju111:venus:mhia06:mhia06:student EDITION:CH 11: CHAPTER Additional Topics in Analtic Geometr C ANALYTIC geometr is the stud of geometric objects using algebraic techniques. René Descartes ( ), the rench philosopher-mathematician, is generall recognized as the founder of the subject. In Chapter, we used analtic geometr to obtain equations of lines. In this chapter, we take a similar approach to the stud of parabolas, ellipses, and hperbolas. Each of these geometric objects is a conic section, that is, the intersection of a plane and a cone. We will derive equations for the conic sections, solve sstems involving equations of conic sections, and eplore a wealth of applications in architecture, communications, engineering, medicine, optics, and space science. 11 OUTLINE 11-1 Conic Sections; Parabola 11- Ellipse 11-3 Hperbola 11-4 Translation and Rotation of Aes 11- Sstems of Nonlinear Equations Chapter 11 Review Chapter 11 Group Activit: ocal Chords Cumulative Review Chapters 10 and 11

2 96 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY 11-1 Conic Sections; Parabola Z Conic Sections Z Defining a Parabola Z Drawing a Parabola Z Standard Equations of Parabolas and Their Graphs Z Applications In this section, we introduce the general concept of a conic section and then discuss the particular conic section called a parabola. In the net two sections, we will discuss two other conic sections called ellipses and hperbolas. Z Conic Sections In Section -1 we found that the graph of a first-degree equation in two variables, A B C (1) where A and B are not both 0, is a straight line, and ever straight line in a rectangular coordinate sstem has an equation of this form. What kind of graph will a seconddegree equation in two variables, A B C D E 0 () where A, B, and C are not all 0, ield for different sets of values of the coefficients? The graphs of equation () for various choices of the coefficients are plane curves obtainable b intersecting a cone* with a plane, as shown in igure 1. These curves are called conic sections. Z igure 1 Conic sections. L V Constant Nappe Circle Ellipse Parabola Hperbola *Starting with a fied line L and a fied point V on L, the surface formed b all straight lines through V making a constant angle with L is called a right circular cone. The fied line L is called the ais of the cone, and V is its verte. The two parts of the cone separated b the verte are called nappes.

3 SECTION 11 1 Conic Sections; Parabola 963 If a plane cuts clear through one nappe, then the intersection curve is called a circle if the plane is perpendicular to the ais and an ellipse if the plane is not perpendicular to the ais. If a plane cuts onl one nappe, but does not cut clear through, then the intersection curve is called a parabola. inall, if a plane cuts through both nappes, but not through the verte, the resulting intersection curve is called a hperbola. A plane passing through the verte of the cone produces a degenerate conic a point, a line, or a pair of lines. Conic sections are ver useful and are readil observed in our immediate surroundings: wheels (circle), the path of water from a garden hose (parabola), some serving platters (ellipses), and the shadow on a wall from a light surrounded b a clindrical or conical lamp shade (hperbola) are some eamples (ig. ). We will discuss man applications of conics throughout the remainder of this chapter. Z igure Eamples of conics. Wheel (circle) (a) Water from garden hose (parabola) (b) Serving platter (ellipse) (c) Lamp light shadow (hperbola) (d) A definition of a conic section that does not depend on the coordinates of points in an coordinate sstem is called a coordinate-free definition. In Appendi A, Section A-3 we gave a coordinate-free definition of a circle and developed its standard equation in a rectangular coordinate sstem. In this and the net two sections, we will give coordinate-free definitions of a parabola, ellipse, and hperbola, and we will develop standard equations for each of these conics in a rectangular coordinate sstem. Z Defining a Parabola The following definition of a parabola does not depend on the coordinates of points in an coordinate sstem: Z DEINITION 1 Parabola A parabola is the set of all points in a plane equidistant from a fied point and a fied line L in the plane. The fied point is called the focus, and the fied line L is called the directri. A line through the focus perpendicular to the directri is called the ais, and the point on the ais halfwa between the directri and focus is called the verte. L P d 1 d d 1 d Ais V(Verte) (ocus) Parabola Directri

4 964 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY ZZZ EXPLORE-DISCUSS 1 In a plane, the reflection of a point P through a line M is the point P such that line M is the perpendicular bisector of the segment PP. The figure shown here can be used to verif that the graph of a parabola is smmetric with respect to line M. (A) Use the figure to show that d d. L d 1 V P c d c M (B) Use the figure and part A to show that d 1 d 1. Can ou now conclude that the graph of a parabola is, in fact, smmetric with respect to its ais of smmetr? Eplain. Z Drawing a Parabola Using Definition 1, we can draw a parabola with fairl simple equipment a straightedge, a right-angle drawing triangle, a piece of string, a thumbtack, and a pencil. Referring to igure 3, tape the straightedge along the line AB and place the thumbtack above the line AB. Place one leg of the triangle along the straightedge as indicated, then take a piece of string the same length as the other leg, tie one end to the thumbtack, and fasten the other end with tape at C on the triangle. Now press the string to the edge of the triangle, and keeping the string taut, slide the triangle along the straightedge. Because DE will alwas equal D, the resulting curve will be part of a parabola with directri AB ling along the straightedge and focus at the thumbtack. Z igure 3 Drawing a parabola. C String D A E B ZZZ EXPLORE-DISCUSS The line through the focus that is perpendicular to the ais of a parabola intersects the parabola in two points G and H. Eplain wh the distance from G to H is twice the distance from to the directri of the parabola.

5 SECTION 11 1 Conic Sections; Parabola 96 Z Standard Equations and Their Graphs Using the definition of a parabola and the distance-between-two-points formula d ( 1 ) ( 1 ) (3) we can derive simple standard equations for a parabola located in a rectangular coordinate sstem with its verte at the origin and its ais along a coordinate ais. We start with the ais of the parabola along the ais and the focus at (a, 0). We locate the parabola in a coordinate sstem as in igure 4 and label ke lines and points. This is an important step in finding an equation of a geometric figure in a coordinate sstem. Note that the parabola opens to the right if a 7 0 and to the left if a 6 0. The verte is at the origin, the directri is a, and the coordinates of M are (a, ). Z igure 4 Parabola with verte at the origin and ais of smmetr the ais. d 1 M (a, ) P (, ) P (, ) d 1 M (a, ) a Directri a d ocus (a, 0) d ocus (a, 0) a Directri a a 0. focus on positive ais (a) a 0. focus on negative ais (b) The point P (, ) is a point on the parabola if and onl if d 1 d d(p, M) d(p, ) ( a) ( ) ( a) ( 0) ( a) ( a) a a a a 4a Use equation (3). Square both sides. Simplif. (4) Equation (4) is the standard equation of a parabola with verte at the origin, ais of smmetr the ais, and focus at (a, 0). B a similar derivation (see Problem 1 in the eercises), the standard equation of a parabola with verte at the origin, ais of smmetr the ais, and focus at (0, a) is given b equation (). 4a () Looking at igure, note that the parabola opens upward if a 7 0 and downward if a 6 0.

6 966 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Z igure Parabola with verte at the origin and ais of smmetr the ais. Directri a (0, a) ocus a d d 1 N (, a) P (, ) N (, a) d 1 P (, ) a d Directri a (0, a) ocus a 0, focus on positive ais (a) a 0, focus on negative ais (b) We summarize these results for eas reference in Theorem 1. Z THEOREM 1 Standard Equations of a Parabola with Verte at (0, 0) 1. 4a Verte: (0, 0) ocus: (a, 0) Directri: a Smmetric with respect to the ais Ais of smmetr the ais. 4a Verte: (0, 0) ocus: (0, a) Directri: a Smmetric with respect to the ais Ais of smmetr the ais 0 a 0 (opens left) 0 a 0 (opens down) 0 a 0 (opens right) 0 a 0 (opens up) EXAMPLE 1 Graphing a Parabola Locate the focus and directri and sketch the graph of 16. SOLUTIONS The equation 16 has the form 4a with 4a 16, so a 4. Therefore, the focus is (4, 0) and the directri is the line 4.

7 SECTION 11 1 Conic Sections; Parabola 967 Hand-Drawn Solution To sketch the graph, we choose some values of that make the right side of the equation a perfect square and solve for Note that must be greater than or equal to 0 for to be a real number. Then we plot the resulting points. Because a 7 0, the parabola opens to the right (ig. 6). Directri 4 10 ocus (4, 0) Graphical Solution To graph 16 on a graphing calculator, we solve this equation for Take the square root of both sides. This results in two functions, 41 and 41. Entering these functions in a graphing utilit (ig. 7) and graphing in a standard viewing window produces the graph of the parabola (ig. 8). 10 Directri Z igure 6 Z igure 7 Z igure 8 MATCHED PROBLEM 1 10 ocus (4, 0) Graph 8, and locate the focus and directri. ZZZ CAUTION ZZZ A common error in making a quick sketch of 4a or 4a is to sketch the first with the ais as its ais of smmetr and the second with the ais as its ais of smmetr. The graph of 4a is smmetric with respect to the ais, and the graph of 4a is smmetric with respect to the ais, as a quick smmetr check will reveal. EXAMPLE inding the Equation of a Parabola (A) ind the equation of a parabola having the origin as its verte, the ais as its ais of smmetr, and (10, ) on its graph. (B) ind the coordinates of its focus and the equation of its directri.

8 968 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY SOLUTIONS (A) Because the ais of smmetr of the parabola is the ais, the parabola has an equation of the form 4a. Because (10, ) is on the graph, we have 4a (10) 4a() 100 0a a Substitute 10 and. Simplif. Divide both sides b 0. Therefore the equation of the parabola is 4() 0 (B) ocus: (0, a) (0, ) Directri: a MATCHED PROBLEM Remark B the graph transformations of Section 1-4, the graph of ( h) k 4a is the same as the graph of 4a shifted h units to the right and k units upward. Solving each equation for the square, we see that the graph of ( h) 4a( k) is the same as the graph of 4a shifted h units to the right and k units upward. So ( h) 4a( k) is the standard equation for a parabola with verte (h, k) and ais of smmetr h. Similarl, ( k) 4a( h) is the standard equation for a parabola with verte (h, k) and ais of smmetr k. In applications of parabolas, we normall choose a coordinate sstem so that the verte of the parabola is the origin and the ais of smmetr is one of the coordinate aes. With such a choice, the equation of the parabola will have one of the standard forms of Theorem 1. (A) ind the equation of a parabola having the origin as its verte, the ais as its ais of smmetr, and (4, 8) on its graph. (B) ind the coordinates of its focus and the equation of its directri. Z Applications Parabolic forms are frequentl encountered in the phsical world. Suspension bridges, arch bridges, microphones, smphon shells, satellite antennas, radio and optical telescopes, radar equipment, solar furnaces, and searchlights are onl a few of man items that use parabolic forms in their design. igure 9(a) illustrates a parabolic reflector used in all reflecting telescopes from 3- to 6-inch home tpes to the 00-inch research instrument on Mount Palomar in California. Parallel light ras from distant celestial bodies are reflected to the focus off a parabolic mirror. If the light source is the sun, then the parallel ras are focused at and we have a solar furnace. Temperatures of over 6,000C have been achieved b such furnaces. If we locate a light source at, then the ras in igure 9(a) reverse, and we have a spotlight or a searchlight. Automobile headlights can use parabolic reflectors with special lenses over the light to diffuse the ras into useful patterns. igure 9(b) shows a suspension bridge, such as the Golden Gate Bridge in San rancisco. The suspension cable is a parabola. It is interesting to note that a freehanging cable, such as a telephone line, does not form a parabola. It forms another curve called a catenar.

9 SECTION 11 1 Conic Sections; Parabola 969 Parallel light ras Parabola Parabola Parabolic reflector (a) Suspension bridge (b) Arch bridge (c) Z igure 9 Uses of parabolic forms. igure 9(c) shows a concrete arch bridge. If all the loads on the arch are to be compression loads (concrete works ver well under compression), then using phsics and advanced mathematics, it can be shown that the arch must be parabolic. EXAMPLE 3 Parabolic Reflector A paraboloid is formed b revolving a parabola about its ais of smmetr. A spotlight in the form of a paraboloid inches deep has its focus inches from the verte. ind, to one decimal place, the radius R of the opening of the spotlight. SOLUTION Step 1. Locate a parabolic cross section containing the ais of smmetr in a rectangular coordinate sstem, and label all known parts and parts to be found. This is a ver important step and can be done in infinitel man was. We can make things simpler for ourselves b locating the verte at the origin and choosing a coordinate ais as the ais of smmetr. We choose the ais as the ais of smmetr of the parabola with the parabola opening upward (ig. 10). R (R, ) (0, ) Spotlight Z igure 10 Step. ind the equation of the parabola in the figure. Because the parabola has the ais as its ais of smmetr and the verte at the origin, the equation is of the form 4a

10 970 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY We are given (0, a) (0, ); thus, a, and the equation of the parabola is 8 Step 3. Use the equation found in step to find the radius R of the opening. Because (R, ) is on the parabola, we have R 8() R inches MATCHED PROBLEM 3 Repeat Eample 3 with a paraboloid 1 inches deep and a focus 9 inches from the verte. ANSWERS TO MATCHED PROBLEMS 1. ocus: (, 0) Directri: (, 0) Directri. (A) 16 (B) ocus: (4, 0); Directri: 4 3. R 0.8 inches 11-1 Eercises 1. Use the geometric objects cone and plane to eplain the difference between a circle and an ellipse.. Use the geometric objects cone and plane to eplain wh a parabola is a single curve, while a hperbola consists of two separate curves. 3. What is a degenerate conic? 4. What is a parabolic mirror?. What happens when light ras parallel to the ais of a parabolic mirror hit the mirror?

11 SECTION 11 1 Conic Sections; Parabola What happens when light ras emitted from the focus of a parabolic mirror hit the mirror? In Problems 7 18, graph each equation, and locate the focus and directri ind the coordinates to two decimal places of the focus for each parabola in Problems In Problems 3, find the equation of a parabola with verte at the origin, ais of smmetr the or ais, and. Directri 3 6. Directri 4 7. ocus (0, 7) 8. ocus (0, ) 9. Directri Directri ocus (, 0) 3. ocus (4, 0) In Problems 33 38, find the equation of the parabola having its verte at the origin, its ais of smmetr as indicated, and passing through the indicated point. 33. ais; (4, ) 34. ais; (4, 8) 3. ais; (3, 6) 36. ais; (, 10) 37. ais; (6, 9) 38. ais; (6, 1) In Problems 39 4, find the first-quadrant points of intersection for each pair of parabolas to three decimal places Consider the parabola with equation 4a. (A) How man lines through (0, 0) intersect the parabola in eactl one point? ind their equations. (B) ind the coordinates of all points of intersection of the parabola with the line through (0, 0) having slope m ind the coordinates of all points of intersection of the parabola with equation 4a and the parabola with equation 4b. 4. The line segment AB through the focus in the figure is called a focal chord of the parabola. ind the coordinates of A and B. 46. The line segment AB through the focus in the figure is called a focal chord of the parabola. ind the coordinates of A and B. (a, 0) In Problems 47 0, use the definition of a parabola and the distance formula to find the equation of a parabola with 47. Directri 4and focus (, ) 48. Directri and focus (3, 6) 49. Directri and focus (6, 4) 0. Directri 3and focus (1, 4) A (0, a) 0 0 4a B A 4a 1. Use the definition of a parabola and the distance formula to derive the equation of a parabola with focus (0, a) and directri afor a 0.. Let be a fied point and let L be a fied line in the plane that contains. Describe the set of all points in the plane that are equidistant from and L. B

12 97 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY APPLICATIONS 3. ENGINEERING The parabolic arch in the concrete bridge in the figure must have a clearance of 0 feet above the water and span a distance of 00 feet. ind the equation of the parabola after inserting a coordinate sstem with the origin at the verte of the parabola and the vertical ais (pointing upward) along the ais of smmetr of the parabola. ocus 00 ft Radiotelescope 100 ft 4. ASTRONOMY The cross section of a parabolic reflector with 6-inch diameter is ground so that its verte is 0.1 inch below the rim (see the figure). 6 inches 0.1 inch (A) ind the equation of the parabola using the ais of smmetr of the parabola as the ais (up positive) and verte at the origin. (B) Determine the depth of the parabolic reflector. 6. SIGNAL LIGHT A signal light on a ship is a spotlight with parallel reflected light ras (see the figure). Suppose the parabolic reflector is 1 inches in diameter and the light source is located at the focus, which is 1. inches from the verte. Signal light Parabolic reflector (A) ind the equation of the parabola after inserting an coordinate sstem with the verte at the origin and the ais (pointing upward) the ais of smmetr of the parabola. (B) How far is the focus from the verte?. SPACE SCIENCE A designer of a 00-foot-diameter parabolic electromagnetic antenna for tracking space probes wants to place the focus 100 feet above the verte (see the figure). ocus (A) ind the equation of the parabola using the ais of smmetr of the parabola as the ais (right positive) and verte at the origin. (B) Determine the depth of the parabolic reflector.

13 SECTION 11 Ellipse Ellipse Z Defining an Ellipse Z Drawing an Ellipse Z Standard Equations of Ellipses and Their Graphs Z Applications We start our discussion of the ellipse with a coordinate-free definition. Using this definition, we show how an ellipse can be drawn and we derive standard equations for ellipses speciall located in a rectangular coordinate sstem. Z Defining an Ellipse The following is a coordinate-free definition of an ellipse: Z DEINITION 1 Ellipse An ellipse is the set of all points P in a plane such that the sum of the distances from P to two distinct fied points in the plane is constant (the constant is required to be greater than the distance between the two fied points). Each of the fied points, and, is called a focus, and together the are called foci. Referring to the figure, the line segment V V through the foci is the major ais. The perpendicular bisector B B of the major ais is the minor ais. Each end of the major ais, V and V, is called a verte. The midpoint of the line segment is called the center of the ellipse. V d 1 d Constant B d 1 P d B V Z Drawing an Ellipse An ellipse is eas to draw. All ou need is a piece of string, two thumbtacks, and a pencil or pen (ig. 1). Place the two thumbtacks in a piece of cardboard. These form the foci of the ellipse. Take a piece of string longer than the distance between the two thumbtacks this represents the constant in the definition and tie each end to a thumbtack.

14 974 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY inall, catch the tip of a pencil under the string and move it while keeping the string taut. The resulting figure is b definition an ellipse. Ellipses of different shapes result, depending on the placement of thumbtacks and the length of the string joining them. Z igure 1 Drawing an ellipse. Note that d 1 d alwas adds up to the length of the string, which does not change. P d 1 d ocus String ocus Z Standard Equations of Ellipses and Their Graphs Using the definition of an ellipse and the distance-between-two-points formula, we can derive standard equations for an ellipse located in a rectangular coordinate sstem. We start b placing an ellipse in the coordinate sstem with the foci on the ais at (c, 0) and (c, 0) with c 7 0 (ig. ). B Definition 1 the constant sum d 1 d is required to be greater than c (the distance between and ). Therefore, the ellipse intersects the ais at points V (a, 0) and V (a, 0) with a 7 c 7 0, and it intersects the ais at points B (b, 0) and B (b, 0) with b 7 0. Z igure on ais. Ellipse with foci b P (, ) d 1 d a 0 (c, 0) (c, 0) a b d 1 d Constant d(, ) c 0 Stud igure : Note first that if P (a, 0), then d 1 d a. (Wh?) Therefore, the constant sum d 1 d is equal to the distance between the vertices. Second, if P (0, b), then d and a b c 1 d a b the Pthagorean theorem; in particular, a 7 b. Referring again to igure, the point P (, ) is on the ellipse if and onl if d 1 d a Using the distance formula for d 1 and d, eliminating radicals, and simplifing (see Problem 41 in the eercises), we obtain the equation of the ellipse pictured in igure : a b 1

15 SECTION 11 Ellipse 97 B similar reasoning (see Problem 4 in Eercises 11-) we obtain the equation of an ellipse centered at the origin with foci on the ais. Both cases are summarized in Theorem 1. Z THEOREM 1 Standard Equations of an Ellipse with Center at (0, 0) 1. a 7 b 7 0 a b 1 intercepts: a (vertices) intercepts: b oci: (c, 0), (c, 0) Major ais length a Minor ais length b c a b. a 7 b 7 0 b a 1 intercepts: b intercepts: a (vertices) oci: (0, c), (0, c) c a b Major ais length a Minor ais length b [Note: Both graphs are smmetric with respect to the ais, ais, and origin. Also, the major ais is alwas longer than the minor ais.] a b a c c a b c 0 c a b a a 0 b ZZZ EXPLORE-DISCUSS 1 The line through a focus of an ellipse that is perpendicular to the major ais intersects the ellipse in two points G and H. or each of the two standard equations of an ellipse with center (0, 0), find an epression in terms of a and b for the distance from G to H. EXAMPLE 1 Graphing an Ellipse ind the coordinates of the foci, find the lengths of the major and minor aes, and graph the following equation:

16 976 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY SOLUTIONS irst, write the equation in standard form b dividing both sides b 144 and determine a and b: * Divide both sides b 144. Simplif a 4 and b 3 intercepts: 4 Major ais length: (4) 8 intercepts: 3 Minor ais length: (3) 6 oci: c a b Substitute a 4 and b c 17 c must be positive. Thus, the foci are (17, 0) and (17, 0). Hand-Drawn Solution Plot the foci and intercepts and sketch the ellipse (ig. 3) 3 Graphical Solution Solve the original equation for : (144 9 )/16 (144 9 )/16 Subtract 9 from both sides. Then divide both sides b 16. Take the square root of both sides. 4 c c 4 This produces the two functions whose graphs are shown in igure 4. Notice that we used a squared viewing window to avoid distorting the shape of the ellipse. Also note the gaps in the graph at 4. This is due to the relativel low resolution of a graphing utilit screen. Z igure Z igure 4 *The dashed boes think boes are used to enclose steps that ma be performed mentall.

17 SECTION 11 Ellipse 977 MATCHED PROBLEM 1 ind the coordinates of the foci, find the lengths of the major and minor aes, and graph the following equation: 4 4 EXAMPLE Graphing an Ellipse ind the coordinates of the foci, find the lengths of the major and minor aes, and graph the following equation: 10 SOLUTION irst, write the equation is standard form b dividing both sides b 10 and determine a and b: Divide both sides b 10. Simplif. a 110 and b 1 intercepts: intercepts: 1.4 oci: c a b 10 c 1 Major ais length: Minor ais length: Substitute a 110, b 1. c must be positive. Thus, the foci are (0, 1) and (0, 1).

18 978 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Hand-Drawn Solution Plot the foci and intercepts and sketch the ellipse (ig. ). 10 c c Graphical Solution Solve for : Subtract from both sides. Take the square root of both sides. Graph and in a squared viewing window (ig. 6). 4 Z igure MATCHED PROBLEM Z igure 6 ind the coordinates of the foci, find the lengths of the major and minor aes, and graph the following equation: 3 18 EXAMPLE 3 inding the Equation of an Ellipse ind an equation of an ellipse in the form M N 1 M, N 7 0 if the center is at the origin, the major ais is along the ais, and (A) (B) Length of major ais 0 Length of minor ais 1 Length of major ais 10 Distance of foci from center 4

19 SECTION 11 Ellipse SOLUTIONS (A) Compute and intercepts and make a rough sketch of the ellipse, as shown in igure b a 1 10 a 0 10 b 1 6 Z igure (B) Make a rough sketch of the ellipse, as shown in igure 8; locate the foci and intercepts, then determine the intercepts using the fact that a b c : b Z igure b b a 1 a b b 3 MATCHED PROBLEM 3 Remark Using graph transformations from Section 1-4, the graphs of ( h) and a ( h) b ( k) 1 b ( k) 1 a a 7 b 7 0 (1) a 7 b 7 0 () ind an equation of an ellipse in the form if the center is at the origin, the major ais is along the ais, and (A) Length of major ais 0 Length of minor ais 30 M N 1 M, N 7 0 (B) Length of minor ais 16 Distance of foci from center 6 are the same as the graphs of Theorem 1 shifted h units to the right and k units upward. Equations (1) and () are therefore the standard forms for equations of ellipses with centers (h, k) and major aes parallel to the ais or ais, respectivel. In applications of ellipses we normall choose a coordinate sstem so that the center of the ellipse is the origin and the major ais lies on one of the coordinate aes. With such a choice, the equation of the ellipse will have one of the standard forms of Theorem 1. ZZZ EXPLORE-DISCUSS (A) Is a circle a special case of an ellipse? Before ou answer, review the coordinate-free definition of an ellipse in this section and the coordinate-free definition of a circle in Appendi A, Section A-3. (B) Wh did we require a 7 b in Theorem 1? (C) State a theorem similar to Theorem 1 for circles.

20 980 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Z Applications Elliptical forms have man application: orbits of satellites, planets, and comets; shapes of galaies; gears and cams; some airplane wings, boat keels, and rudders; tabletops; public fountains; and domes in buildings are a few eamples (ig. 9). Planet Sun Planetar motion Elliptical gears Elliptical dome (a) (b) (c) Z igure 9 forms. Uses of elliptical Johannes Kepler ( ), a German astronomer, discovered that planets move in elliptical orbits, with the sun at a focus, and not in circular orbits as had been thought before [ig. 9(a)]. igure 9(b) shows a pair of elliptical gears with pivot points at foci. Such gears transfer constant rotational speed to variable rotational speed, and vice versa. igure 9(c) shows an elliptical dome. An interesting propert of such a dome is that a sound or light source at one focus will reflect off the dome and pass through the other focus. One of the chambers in the Capitol Building in Washington, D.C., has such a dome, and is referred to as a whispering room because a whispered sound at one focus can be easil heard at the other focus. A fairl recent application in medicine is the use of elliptical reflectors and ultrasound to break up kidne stones. A device called a lithotripter is used to generate intense sound waves that break up the stone from outside the bod, thus avoiding surger. To be certain that the waves do not damage other parts of the bod, the reflecting propert of the ellipse is used to design and correctl position the lithotripter. EXAMPLE 4 Medicinal Lithotrips A lithotripter is formed b rotating the portion of an ellipse below the minor ais around the major ais (ig. 10). The lithotripter is 0 centimeters wide and 16 centimeters deep. If the ultrasound source is positioned at one focus of the ellipse and the kidne stone at the other, then all the sound waves will pass through the kidne stone. How far from the kidne stone should the point V on the base of the lithotripter be positioned to focus the sound waves on the kidne stone? Round the answer to one decimal place.

21 SECTION 11 Ellipse 981 Ultrasound source Kidne stone Base V 0 cm 16 cm Z igure 10 Lithotripter. SOLUTION rom igure 10 we see that a 16 and b 10 for the ellipse used to form the lithotripter. Thus, the distance c from the center to either the kidne stone or the ultrasound source is given b c a b and the distance from the base of the lithotripter to the kidne stone is centimeters. MATCHED PROBLEM 4 Because lithotrips is an eternal procedure, the lithotripter described in Eample 4 can be used onl on stones within 1. centimeters of the surface of the bod. Suppose a kidne stone is located 14 centimeters from the surface. If the diameter is kept fied at 0 centimeters, how deep must a lithotripter be to focus on this kidne stone? Round answer to one decimal place. ANSWERS TO MATCHED PROBLEMS 1. 1 oci: (3, 0), (3, 0) Major ais length 4 Minor ais length 0 1

22 98 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. 18 oci: (0, 1), (0, 1) Major ais length Minor ais length (A) (B) centimeters Eercises 1. Does the graph of an ellipse pass the vertical line test (Section 1-)? Eplain.. Wh are two equations required to graph an ellipse on a graphing calculator? 3. Does the graph of either equation in Problem pass the horizontal line test (Section 1-6)? Eplain. 4. Given the and intercepts of an ellipse centered at the origin, describe a procedure for sketching the graph of the ellipse.. Repeat Problem 4 for a circle. 6. Some sa that the distinction between an ellipse and a circle is a distinction without a difference. Do ou agree or disagree? Wh? In Problems 7 1, sketch a graph of each equation, find the coordinates of the foci, and find the lengths of the major and minor aes In Problems 13 16, match each equation with one of graphs (a) (d) (a) (c) In Problems 17, sketch a graph of each equation, find the coordinates of the foci, and find the lengths of the major and minor aes (b) (d)

23 SECTION 11 Ellipse 983 In Problems 3 34, find an equation of an ellipse in the form if the center is at the origin, and 3. The graph is 4. The graph is 10. The graph is 6. The graph is M N Major ais on ais 8. Major ais on ais Major ais length 10 Major ais length 14 Minor ais length 6 Minor ais length Major ais on ais 30. Major ais on ais Major ais length Major ais length 4 Minor ais length 16 Minor ais length Major ais on ais Major ais length 16 Distance of foci from center 6 3. Major ais on ais Major ais length 4 Distance of foci from center Major ais on ais Minor ais length 0 Distance of foci from center Major ais on ais Minor ais length 14 Distance of foci from center M, N Eplain wh an equation whose graph is an ellipse does not define a function. 36. Consider all ellipses having (0, 1) as the ends of the minor ais. Describe the connection between the elongation of the ellipse and the distance from a focus to the origin ind an equation of the set of points in a plane, each of whose distance from (, 0) is one-half its distance from the line 8. Identif the geometric figure. 38. ind an equation of the set of points in a plane, each of whose distance from (0, 9) is three-fourths its distance from the line 16. Identif the geometric figure. 39. Let and be two points in the plane and let c denote the constant d(, ). Describe the set of all points P in the plane such that the sum of the distances from P to and is equal to the constant c. 40. Let and be two points in the plane and let c be a constant such that 0 6 c 6 d(, ). Describe the set of all points P in the plane such that the sum of the distances from P to and is equal to the constant c. 41. Stud the following derivation of the standard equation of an ellipse with foci (c, 0), intercepts (a, 0), and intercepts (0, b). Eplain wh each equation follows from the equation that precedes it. [Hint: Recall from igure that a b c.] d 1 d a ( c) a ( c) ( c) 4a 4a( c) ( c) ( c) a c a ( c) a c c a1 c a b a c a b 1 4. Stud the following derivation of the standard equation of an ellipse with foci (0, c), intercepts (0, a), and intercepts (b, 0). Eplain wh each equation follows from the equation that precedes it. [Hint: Recall from igure that a b c.] d 1 d a a1 c a b a c b a 1 a ( c) a ( c) ( c) 4a 4a ( c) ( c) ( c) a c a ( c) a c c a

24 984 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY APPLICATIONS 43. ENGINEERING The semielliptical arch in the concrete bridge in the figure must have a clearance of 1 feet above the water and span a distance of 40 feet. ind the equation of the ellipse after inserting a coordinate sstem with the center of the ellipse at the origin and the major ais on the ais. The ais points up, and the ais points to the right. How much clearance above the water is there feet from the bank? Elliptical bridge (A) If the straight-line leading edge is parallel to the major ais of the ellipse and is 1.14 feet in front of it, and if the leading edge is 46.0 feet long (including the width of the fuselage), find the equation of the ellipse. Let the ais lie along the major ais (positive right), and let the ais lie along the minor ais (positive forward). (B) How wide is the wing in the center of the fuselage (assuming the wing passes through the fuselage)? Compute quantities to three significant digits. 46. NAVAL ARCHITECTURE Currentl, man high-performance racing sailboats use elliptical keels, rudders, and main sails for the reasons stated in Problem 4 less drag along the trailing edge. In the accompaning figure, the ellipse containing the keel has a 1.0-foot major ais. The straight-line leading edge is parallel to the major ais of the ellipse and 1.00 foot in front of it. The chord is 1.00 foot shorter than the major ais. 44. DESIGN A 4 8 foot elliptical tabletop is to be cut out of a 4 8 foot rectangular sheet of teak plwood (see the figure). To draw the ellipse on the plwood, how far should the foci be located from each edge and how long a piece of string must be fastened to each focus to produce the ellipse (see ig. 1 in the tet)? Compute the answer to two decimal places. String Elliptical table 4. AERONAUTICAL ENGINEERING Of all possible wing shapes, it has been determined that the one with the least drag along the trailing edge is an ellipse. The leading edge ma be a straight line, as shown in the figure. One of the most famous planes with this design was the World War II British Spitfire. The plane in the figure has a wingspan of 48.0 feet. Leading edge Rudder Keel (A) ind the equation of the ellipse. Let the ais lie along the minor ais of the ellipse, and let the ais lie along the major ais, both with positive direction upward. (B) What is the width of the keel, measured perpendicular to the major ais, 1 foot up the major ais from the bottom end of the keel? Compute quantities to three significant digits. Elliptical wings and tail uselage Trailing edge

25 SECTION 11 3 Hperbola Hperbola Z Defining a Hperbola Z Drawing a Hperbola Z Standard Equations of Hperbolas and Their Graphs Z Applications As before, we start with a coordinate-free definition of a hperbola. Using this definition, we show how a hperbola can be drawn and we derive standard equations for hperbolas speciall located in a rectangular coordinate sstem. Z Defining a Hperbola The following is a coordinate-free definition of a hperbola: Z DEINITION 1 Hperbola A hperbola is the set of all points P in a plane such that the absolute value of the difference of the distances of P to two distinct fied points in the plane is a positive constant d 1 d Constant P d 1 d (the constant is required to be less than the V distance between the two fied points). Each of the fied points, and, is called a focus. V The intersection points V and V of the line through the foci and the two branches of the hperbola are called vertices, and each is called a verte. The line segment V V is called the transverse ais. The midpoint of the transverse ais is the center of the hperbola. Z Drawing a Hperbola Thumbtacks, a straightedge, string, and a pencil are all that are needed to draw a hperbola (ig. 1). Place two thumbtacks in a piece of cardboard these form the foci of the hperbola. Rest one corner of the straightedge at the focus so that it is free to rotate about this point. Cut a piece of string shorter than the length of the straightedge, and fasten one end to the straightedge corner A and the other end to the thumbtack at. Now push the string with a pencil up against the straightedge at B. Keeping the string taut, rotate the straightedge about, keeping the corner at. The resulting curve will be part of a hperbola. Other parts of the hperbola can be drawn b changing the

26 986 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY position of the straightedge and string. To see that the resulting curve meets the conditions of the definition, note that the difference of the distances B and B is B B B BA B BA A (B BA) a Straightedge b a String length length b Constant B A String Z igure 1 Drawing a hperbola. Z Standard Equations of Hperbolas and Their Graphs Using the definition of a hperbola and the distance-between-two-points formula, we can derive standard equations for a hperbola located in a rectangular coordinate sstem. We start b placing a hperbola in the coordinate sstem with the foci on the ais at (c, 0) and (c, 0) with c 7 0 (ig. ). B Definition 1, the constant difference d 1 d is required to be less than c (the distance between and ). Therefore, the hperbola intersects the ais at points V (a, 0) and V (a, 0) with c 7 a 7 0. The hperbola does not intersect the ais, because the constant difference d 1 d is required to be positive b Definition 1. Z igure Hperbola with foci on the ais. d P (, ) (c, 0) a 1 a d (c, 0) c 0 d 1 d Positive constant d(, )

27 Stud igure : Note that if P (a, 0), then d 1 d a. (Wh?) Therefore the constant d 1 d is equal to the distance between the vertices. It is convenient to let b c a, so that c a b. (Unlike the situation for ellipses, b ma be greater than or equal to a.) Referring again to igure, the point P (, ) is on the hperbola if and onl if d 1 d a Using the distance formula for d 1 and d, eliminating radicals, and simplifing (see Problem 3 in the eercises), we obtain the equation of the hperbola pictured in igure : a b 1 SECTION 11 3 Hperbola 987 Although the hperbola does not intersect the ais, the points (0, b) and (0, b) are significant; the line segment joining them is called the conjugate ais of the hperbola. Note that the conjugate ais is perpendicular to the transverse ais, that is, the line segment joining the vertices (a, 0) and (a, 0). The rectangle with corners (a, b), (a, b), (a, b), and (a, b) is called the asmptote rectangle because its etended diagonals are asmptotes for the hperbola (ig. 3). In other words, the hperbola approaches the lines b a as becomes larger (see Problems 49 and 0 in the eercises). As a result, it is helpful to include the asmptote rectangle and its etended diagonals when sketching the graph of a hperbola. Asmptote Asmptote b a b a b a 0 b a 1 a b Z igure 3 Asmptotes. Note that the four corners of the asmptote rectangle (ig. 3) are equidistant from the origin, at distance a b c. Therefore, A circle, with center at the origin, that passes through all four corners of the asmptote rectangle of a hperbola also passes through its foci. B similar reasoning (see Problem 4 in the eercises) we obtain the equation of a hperbola centered at the origin with foci on the ais. Both cases are summarized in Theorem 1.

28 988 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Z THEOREM 1 Standard Equations of a Hperbola with Center at (0, 0) 1. a b 1 intercepts: a (vertices) intercepts: none oci: (c, 0), (c, 0) c a b c a c c a b b Transverse ais length a Conjugate ais length b Asmptotes: b a. a b 1 intercepts: none intercepts: a (vertices) oci: (0, c), (0, c) b c a c b c a b a c Transverse ais length a Conjugate ais length b Asmptotes: a b [Note: Both graphs are smmetric with respect to the ais, ais, and origin.] ZZZ EXPLORE-DISCUSS 1 The line through a focus of a hperbola that is perpendicular to the transverse ais intersects the hperbola in two points G and H. or each of the two standard equations of a hperbola with center (0, 0), find an epression in terms of a and b for the distance from G to H. EXAMPLE 1 Graphing Hperbolas ind the coordinates of the foci, find the lengths of the transverse and conjugate aes, find the equations of the asmptotes, and graph the following equation:

29 SECTION 11 3 Hperbola 989 SOLUTIONS irst, write the equation in standard form b dividing both sides b 144 and determine a and b: Divide both sides b 144. Simplif a 4 and b 3 intercepts: 4 Transverse ais length (4) 8 intercepts: none Conjugate ais length (3) 6 oci: c a b 16 9 c Substitute a 4 and b 3. Thus, the foci are (, 0) and (, 0). Hand-Drawn Solution Plot the foci and intercepts, sketch the asmptote rectangle and the asmptotes, then sketch the hperbola (ig. 4). The equations of the asmptotes are 3 4 (note that the diagonals of the asmptote rectangle have slope 3 4). c 6 c c 6 Graphical Solution Solve for : (9 144)/16 (9 144)/16 Subtract 9 from both sides. Divide both sides b 16. Take the square root of both sides. This produces the functions 1 (9 144)/16 and (9 144)/16 whose graphs are shown in igure. 6 Z igure Z igure

30 990 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY MATCHED PROBLEM 1 ind the coordinates of the foci, find the lengths of the transverse and conjugate aes, and graph the following equation: EXAMPLE Graphing Hperbolas ind the coordinates of the foci, find the lengths of the transverse and conjugate aes, find the equations of the asmptotes, and graph the following equation: SOLUTIONS Write the equation in standard form: a 3 and b 4 Divide both sides b 144. intercepts: 3 Transverse ais length (3) 6 intercepts: none Conjugate ais length (4) 8 oci: c a b 9 16 c Substitute a 3 and b 4. Thus, the foci are (0, ) and (0, ). Hand-Drawn Solution Plot the foci and intercepts, sketch the asmptote rectangle and the asmptotes, then sketch the hperbola (ig. 6). The equations of the asmptotes are 3 4 (note that the diagonals of the asmptote rectangle have slope 3 4). Graphical Solution Solve for : (144 9 )/16 (144 9 )/16 Add 9 to both sides. Divide both sides b 16. Take the square root of both sides.

31 SECTION 11 3 Hperbola 991 This gives us the functions: 6 6 c c 6 1 (144 9 )/16 (144 9 )/16 whose graphs are shown in igure 7. 6 and c Z igure 6 Z igure 7 6 MATCHED PROBLEM ind the coordinates of the foci, find the lengths of the transverse and conjugate aes, and graph the following equation: Two hperbolas of the form M N 1 and N M 1 M, N 7 0 are called conjugate hperbolas. In Eamples 1 and and in Matched Problems 1 and, the hperbolas are conjugate hperbolas the share the same asmptotes. ZZZ CAUTION ZZZ When making a quick sketch of a hperbola, it is a common error to have the hperbola opening up and down when it should open left and right, or vice versa. The mistake can be avoided if ou first locate the intercepts accuratel. EXAMPLE 3 Graphing Hperbolas ind the coordinates of the foci, find the lengths of the transverse and conjugate aes, and graph the following equation: 10

32 99 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY SOLUTIONS a 1 and Divide both sides b 10. b 110 intercepts: 1 intercepts: none Transverse ais length Conjugate ais length oci: c a b 10 1 c 11 Substitute a 1 and b 110. Thus, the foci are (11, 0) and (11, 0). Hand-Drawn Solution Plot the foci and intercepts, sketch the asmptote rectangle and the asmptotes, then sketch the hperbola (ig. 8). c c c Graphical Solution Solve for : Add and 10 to both sides. Take the square root of both sides. This gives us two functions, 1 10 and 10, which are graphed in igure Z igure 8 6 Z igure 9 MATCHED PROBLEM 3 ind the coordinates of the foci, find the lengths of the transverse and conjugate aes, and graph the following equation: 3 1

33 SECTION 11 3 Hperbola 993 EXAMPLE 4 inding the Equation of a Hperbola ind an equation of a hperbola in the form M N 1 M, N 7 0 if the center is at the origin, and: (A) Length of transverse ais is 1 (B) Length of transverse ais is 6 Length of conjugate ais is 0 Distance of foci from center is SOLUTIONS (A) Start with and find a and b: a b 1 a 1 6 and b 0 10 Thus, the equation is (B) Start with a b 1 and find a and b: a 6 3 b b 3 To find b, sketch the asmptote rectangle (ig. 10), label known parts, and use the Pthagorean theorem: b 3 16 b 4 Z igure 10 Asmptote rectangle. Thus, the equation is

34 994 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY MATCHED PROBLEM 4 ind an equation of a hperbola in the form M N 1 M, N 7 0 if the center is at the origin, and: (A) Length of transverse ais is 0 (B) Length of conjugate ais is 1 Length of conjugate ais is 30 Distance of foci from center is 9 ZZZ EXPLORE-DISCUSS (A) Does the line with equation intersect the hperbola with equation ( /4) 1? If so, find the coordinates of all intersection points. (B) Does the line with equation 3 intersect the hperbola with equation ( /4) 1? If so, find the coordinates of all intersection points. (C) or which values of m does the line with equation m intersect the hperbola? ind the coordinates of all intersection points. a b 1 Z Applications You ma not be aware of the man important uses of hperbolic forms. The are encountered in the stud of comets; the loran sstem of navigation for pleasure boats, ships, and aircraft; sundials; capillar action; nuclear reactor cooling towers; optical and radio telescopes; and contemporar architectural structures. The TWA building at Kenned Airport is a hperbolic paraboloid, and the St. Louis Science Center Planetarium is a hperboloid. With such structures, thin concrete shells can span large spaces [ig. 11(a)]. Some comets from outer space occasionall enter the sun s gravitational field, follow a hperbolic path around the sun (with the sun as a focus), and then leave, never to be seen again [ig. 11(b)]. Eample illustrates the use of hperbolas in navigation. Z igure 11 forms. Uses of hperbolic Comet Sun St. Louis Planetarium (a) Comet around sun (b)

35 SECTION 11 3 Hperbola 99 EXAMPLE Navigation A ship is traveling on a course parallel to and 60 miles from a straight shoreline. Two transmitting stations, S 1 and S, are located 00 miles apart on the shoreline (ig. 1). B timing radio signals from the stations, the ship s navigator determines that the ship is between the two stations and 0 miles closer to S than to S 1. ind the distance from the ship to each station. Round answers to one decimal place. d 1 60 miles d S 1 S 00 miles Z igure 1 d 1 d 0. SOLUTION 100 Z igure (, 60) S 1 S 100 If d 1 and d are the distances from the ship to S 1 and S, respectivel, then d 1 d 0 and the ship must be on the hperbola with foci at S 1 and S and fied difference 0, as illustrated in igure 13. In the derivation of the equation of a hperbola, we represented the fied difference as a. Thus, for the hperbola in igure 13 we have c 100 The equation for this hperbola is a 1 (0) b ,37 6 9,37 1 Substitute 60 and solve for (see ig. 13): ,37 1 3, , ,600 9,37 9,37 Add 60 9,37 to both sides. Multipl both sides b 6. Simplif.

36 996 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Thus, (The negative square root is discarded, because the ship is closer to S than to S 1.) Distance from ship to S 1 Distance from ship to S d 1 ( ) 60 d ( ) 60 10, , miles 9.6 miles Notice that the difference between these two distances is 0, as it should be. MATCHED PROBLEM Repeat Eample if the ship is 80 miles closer to S than to S 1. Ship S 3 q S 1 S q p 1 1 p Z igure 14 Loran navigation. Eample illustrates a simplified form of the loran (LOng RAnge Navigation) sstem. In practice, three transmitting stations are used to send out signals simultaneousl (ig. 14), instead of the two used in Eample. A computer onboard a ship will record these signals and use them to determine the differences of the distances that the ship is to S 1 and S, and to S and S 3. Plotting all points so that these distances remain constant produces two branches, p 1 and p, of a hperbola with foci S 1 and S, and two branches, q 1 and q, of a hperbola with foci S and S 3. It is eas to tell which branches the ship is on b comparing the signals from each station. The intersection of a branch of each hperbola locates the ship and the computer epresses this in terms of longitude and latitude. ANSWERS TO MATCHED PROBLEMS c 10 c c oci: (41, 0), (41, 0) Transverse ais length 10 Conjugate ais length c c oci: (0, 41), (0, 41) Transverse ais length 8 Conjugate ais length 10 c 10

37 SECTION 11 3 Hperbola c c oci: (0, 4), (0, 4) Transverse ais length Conjugate ais length 4 c 6 4. (A) (B). d miles, d 79. miles Eercises 1. What is the transverse ais of a hperbola and how do ou find it?. What is the conjugate ais of a hperbola and how do ou find it? 3. How do ou find the foci of a hperbola? 4. What is the asmptote rectangle and how is it used to graph a hperbola?. Given the equation a b 1, replace 1 with 0 and then solve for. Discuss how the results can serve as a memor aid when graphing a hperbola. 6. Given the equation a b 1, replace 1 with 0 and then solve for. Discuss how the results can serve as a memor aid when graphing a hperbola. In Problems 7 10, match each equation with one of graphs (a) (d) Sketch a graph of each equation in Problems 11 18, find the coordinates of the foci, and find the lengths of the transverse and conjugate aes (c) Sketch a graph of each equation in Problems 19, find the coordinates of the foci, and find the lengths of the transverse and conjugate aes (d) 9 1 (a) (b)

38 998 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY In Problems 3 34, find an equation of a hperbola in the form or M N 1 N M 1 if the center is at the origin, and: 3. The graph is 4. The graph is 10. The graph is 6. The graph is (, 4) (3, ) 7. Transverse ais on ais Transverse ais length 14 Conjugate ais length Transverse ais on ais Transverse ais length 8 Conjugate ais length 6 9. Transverse ais on ais Transverse ais length 4 Conjugate ais length Transverse ais on ais Transverse ais length 16 Conjugate ais length Transverse ais on ais Transverse ais length 18 Distance of foci from center Transverse ais on ais Transverse ais length 16 Distance of foci from center Conjugate ais on ais Conjugate ais length 14 Distance of foci from center M, N 7 0 (4, ) 10 (, 3) Conjugate ais on ais Conjugate ais length 10 Distance of foci from center 170 In Problems 3 4, find the equations of the asmptotes of each hperbola (A) How man hperbolas have center at (0, 0) and a focus at (1, 0)? ind their equations. (B) How man ellipses have center at (0, 0) and a focus at (1, 0)? ind their equations. (C) How man parabolas have center at (0, 0) and focus at (1, 0)? ind their equations. 44. How man hperbolas have the lines as asmptotes? ind their equations. 4. ind all intersection points of the graph of the hperbola 1 with the graph of each of the following lines: (A) 0. (B) or what values of m will the graph of the hperbola and the graph of the line m intersect? ind the coordinates of these intersection points. 46. ind all intersection points of the graph of the hperbola 1 with the graph of each of the following lines: (A) 0. (B) or what values of m will the graph of the hperbola and the graph of the line m intersect? ind the coordinates of these intersection points. 47. ind all intersection points of the graph of the hperbola 4 1 with the graph of each of the following lines: (A) (B) 3 or what values of m will the graph of the hperbola and the graph of the line m intersect? ind the coordinates of these intersection points. 48. ind all intersection points of the graph of the hperbola 4 1 with the graph of each of the following lines: (A) (B) 3 or what values of m will the graph of the hperbola and the graph of the line m intersect? ind the coordinates of these intersection points. 49. Consider the hperbola with equation a b 1

39 (A) Show that b a1 a. (B) Eplain wh the hperbola approaches the lines b a as becomes larger. (C) Does the hperbola approach its asmptotes from above or below? Eplain. 0. Consider the hperbola with equation a b 1 (A) Show that a b1 b. (B) Eplain wh the hperbola approaches the lines a b as becomes larger. (C) Does the hperbola approach its asmptotes from above or below? Eplain. 1. Let and be two points in the plane and let c be a constant such that c 7 d(, ). Describe the set of all points P in the plane such that the absolute value of the difference of the distances from P to and is equal to the constant c.. Let and be two points in the plane and let c denote the constant d(, ). Describe the set of all points P in the plane such that the absolute value of the difference of the distances from P to and is equal to the constant c. 3. Stud the following derivation of the standard equation of a hperbola with foci (c, 0), intercepts (a, 0), and endpoints of the conjugate ais (0, b). Eplain wh each equation follows from the equation that precedes it. [Hint: Recall that c a b.] d 1 d a ( c) a ( c) ( c) 4a 4a( c) ( c) ( c) a c a ( c) a c c a d 1 d a ( c) a ( c) ( c) 4a 4a ( c) ( c) ( c) a c a ( c) a c c a1 c a b a c a b 1 SECTION 11 3 Hperbola 999 ECCENTRICITY Problems and 6 and Problems 37 and 38 in Eercise 11- are related to a propert of conics called eccentricit, which is denoted b a positive real number E. Parabolas, ellipses, and hperbolas all can be defined in terms of E, a fied point called a focus, and a fied line not containing the focus called a directri as follows: The set of points in a plane each of whose distance from a fied point is E times its distance from a fied line is an ellipse if 0 6 E 6 1, a parabola if E 1, and a hperbola if E ind an equation of the set of points in a plane each of whose distance from (3, 0) is three-halves its distance from the line 4 3. Identif the geometric figure. 6. ind an equation of the set of points in a plane each of whose distance from (0, 4) is four-thirds its distance from the line 9 4. Identif the geometric figure. APPLICATIONS 7. ARCHITECTURE An architect is interested in designing a thin-shelled dome in the shape of a hperbolic paraboloid, as shown in igure (a). ind the equation of the hperbola located in a coordinate sstem [ig. (b)] satisfing the indicated conditions. How far is the hperbola above the verte 6 feet to the right of the verte? Compute the answer to two decimal places. a a1 c a b a c a b 1 4. Stud the following derivation of the standard equation of a hperbola with foci (0, c), intercepts (0, a), and endpoints of the conjugate ais (b, 0). Eplain wh each equation follows from the equation that precedes it. [Hint: Recall that c a b.] Parabola Hperbola Hperbolic paraboloid (a)

40 1000 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY (8, 1) 10 Hperbola part of dome (b) is the radius of the top and the base? What is the radius of the smallest circular cross section in the tower? Compute answers to three significant digits. 9. SPACE SCIENCE In tracking space probes to the outer planets, NASA uses large parabolic reflectors with diameters equal to two-thirds the length of a football field. Needless to sa, man design problems are created b the weight of these reflectors. One weight problem is solved b using a hperbolic reflector sharing the parabola s focus to reflect the incoming electromagnetic waves to the other focus of the hperbola where receiving equipment is installed (see the figure). 8. NUCLEAR POWER A nuclear reactor cooling tower is a hperboloid, that is, a hperbola rotated around its conjugate ais, as shown in igure (a). The equation of the hperbola in igure (b) used to generate the hperboloid is Common focus Hperbola Incoming wave Hperbola focus Receiving cone Parabola (a) Nuclear reactor cooling tower (a) Radio telescope 00 (b) Hperbola part of dome (b) If the tower is 00 feet tall, the top is 10 feet above the center of the hperbola, and the base is 30 feet below the center, what 00 or the receiving antenna shown in the figure, the common focus is located 10 feet above the verte of the parabola, and focus (for the hperbola) is 0 feet above the verte. The verte of the reflecting hperbola is 110 feet above the verte for the parabola. Introduce a coordinate sstem b using the ais of the parabola as the ais (up positive), and let the ais pass through the center of the hperbola (right positive). What is the equation of the reflecting hperbola? Write in terms of.

41 SECTION 11 4 Translation and Rotation of Aes Translation and Rotation of Aes Z Translation of Aes Z Translation Used in Graphing Z Rotation of Aes Z Rotation Used in Graphing Z Identifing Conics In Sections 11-1, 11-, and 11-3 we found standard equations for parabolas, ellipses, and hperbolas with aes on the coordinate aes and centered relative to the origin. Each of those standard equations was a special case of the equation A B C D E 0 (1) for appropriate constants A, B, C, D, E, and. In this section we show that ever equation of the form (1) has a graph that is either a conic, a degenerate conic (that is, a point, a line, or a pair of lines), or the empt set. The difficult is that a conic with an equation of form (1) might not be centered at the origin, and might have aes that are skewed with respect to the coordinate aes. To overcome the difficult we use two basic mathematical tools: translation of aes and rotation of aes. With these tools we will be able to choose a new coordinate sstem (that depends on the constants A, B, C, D, E, and ) in which the equation has an especiall transparent and useful form. 0 (0, 0) (0, 0) (h, k) 0 P Z igure 1 Translation of coordinates. (, ) (, ) Z Translation of Aes If ou move a sheet of paper on a desk top, without rotating the paper and without flipping it over, ou translate the paper to its new position. Similarl, a translation of coordinate aes occurs when the new coordinate aes have the same direction as, and are parallel to, the original coordinate aes. To see how coordinates in the original sstem are changed when moving to the translated sstem, and vice versa, refer to igure 1. A point P in the plane has two sets of coordinates: (, ) in the original sstem and (, ) in the translated sstem. If the coordinates of the origin of the translated sstem are (h, k) relative to the original sstem, then the old and new coordinates are related as given in Theorem 1. Z THEOREM 1 Translation ormulas 1. h. h k k It can be shown that these formulas hold for (h, k) located anwhere in the original coordinate sstem.

42 100 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY EXAMPLE 1 Equation of a Curve in a Translated Sstem A curve has the equation ( 4) ( 1) 36 If the origin is translated to (4, 1), find the equation of the curve in the translated sstem and identif the curve. SOLUTION Because (h, k) (4, 1), use translation formulas to obtain, after substitution, h 4 k 1 36 This is the equation of a circle of radius 6 with center at the new origin. The coordinates of the new origin in the original coordinate sstem are (4, 1) (ig. ). Note that this result agrees with our general treatment of the circle in Section B-3. 0 A (4, 1) 10 Z igure ( 4) ( 1) 36. MATCHED PROBLEM 1 A curve has the equation ( ) 8( 3). If the origin is translated to (3, ), find an equation of the curve in the translated sstem and identif the curve. Suppose the coordinate aes in the sstem have been translated to (h, k), as in igure 1 on page Then, as illustrated b Eample 1, the circle r has the equation ( h) ( k) r in the original sstem. In a similar manner we use the standard equations for the parabola, ellipse, and hperbola centered at the origin to obtain more general standard equations for conics centered at the point (h, k) (see Table 1). Note that when h 0 and k 0 the standard equations of Table 1 are eactl the standard equations obtained in Sections 11-1, 11-, and 11-3.

43 SECTION 11 4 Translation and Rotation of Aes 1003 Table 1 Standard Equations for Conics Parabolas ( h) 4a( k) ( k) 4a( h) a V (h, k) Verte (h, k) ocus (h, k a) a 0 opens up a 0 opens down V (h, k) a Verte (h, k) ocus (h a, k) a 0 opens left a 0 opens right Circles ( h) ( k) r Center (h, k) Radius r r C (h, k) ( h) b ( k) 1 a Ellipses a b 0 ( h) a ( k) 1 b b a (h, k) Center (h, k) Major ais a Minor ais b a Center (h, k) Major ais a Minor ais b (h, k) b Hperbolas (h, k) ( h) b a a ( k) b 1 Center (h, k) Transverse ais a Conjugate ais b (h, k) ( h) a a b ( h) 1 b Center (h, k) Transverse ais a Conjugate ais b

44 1004 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Z Translation Used in Graphing An equation of the form A C D E 0 () has a graph that is a conic, a degenerate conic, or the empt set [note that equation () is the same as equation (1) on page 1001 with B 0]. To see this, we use the technique of completing the square discussed in Section -3. If we can transform equation () into one of the standard forms of Table 1, then we will be able to identit its graph and sketch it rather quickl. Some eamples should help make the process clear. EXAMPLE Graphing a Conic Given the equation = 0 (3) (A) Transform the equation into one of the standard forms in Table 1 and identif the conic. (B) ind the equation in the translated sstem. (C) Graph the conic. SOLUTIONS (A) Complete the square in equation (3) relative to each variable that is squared in this case : ( 3) 4( ) Add 4 1 to both sides. Add 9 to both sides to complete the square on the left side. actor. (4) rom Table 1 we recognize equation (4) as an equation of a parabola opening to the right with verte at (h, k) (, 3). (B) ind the equation of the parabola in the translated sstem with origin 0 at (h, k) (, 3). The equations of translation are read directl from equation (4): 3 Making these substitutions in equation (4) we obtain 4 () the equation of the parabola in the sstem.

45 SECTION 11 4 Translation and Rotation of Aes 100 (C) Graph equation () in the sstem following the process discussed in Section The resulting graph is the graph of the original equation relative to the original coordinate sstem (ig. 3). A (, 3) 0 Z igure 3 MATCHED PROBLEM Repeat Eample for the equation EXAMPLE 3 Graphing a Conic Given the equation = 0 (A) Transform the equation into one of the standard forms in Table 1 and identif the conic. (B) ind the equation in the translated sstem. (C) Graph the conic. (D) ind the coordinates of an foci relative to the original sstem. SOLUTIONS (A) Complete the square relative to both and ( 4 ) 4( 6 ) 36 9( 4 4) 4( 6 9) ( ) 4( 3) 36 ( ) ( 3) Add 36 to both sides. actor out coefficients of and. Complete squares. actor. Divide both sides b 36.

46 1006 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY rom Table 1 we recognize the last equation as an equation of a hperbola opening left and right with center at (h, k) (, 3). (B) ind the equation of the hperbola in the translated sstem with origin 0 at (h, k) = (, 3). The equations of translation are read directl from the last equation in part A: Making these substitutions, we obtain 3 the equation of the hperbola in the sstem (C) Hand-Drawn Solution Graph the equation obtained in part B in the sstem following the process discussed in the last section. The resulting graph is the graph of the original equation relative to the original coordinate sstem (ig. 4). c 10 Z igure 4 c (C) Graphing Calculator Solution To graph the equation of this eample on a graphing calculator, write it as a quadratic equation in the variable, and use the quadratic formula to solve for Write in the form a b c ( ) 0 Use the quadratic formula with a 4, b 4, and c (4)( ) (6) The two functions determined b equation (6) are graphed in igure Z igure (D) ind the coordinates of the foci. To find the coordinates of the foci in the original sstem, first find the coordinates in the translated sstem: c 3 13 c 113 c 113

47 SECTION 11 4 Translation and Rotation of Aes 1007 Thus, the coordinates in the translated sstem are (113, 0) and (113, 0) Now, use h k 3 to obtain (113, 3) and (113, 3) as the coordinates of the foci in the original sstem. MATCHED PROBLEM 3 Repeat Eample 3 for the equation = 0 ZZZ EXPLORE-DISCUSS 1 If A 0 and C 0, show that the translation of aes D A and E transforms the equation A C D E 0 C into an equation of the form A C K. EXAMPLE 4 inding the Equation of a Conic ind the equation of a hperbola with vertices on the line 4, conjugate ais on the line 3, length of the transverse ais 4, and length of the conjugate ais 6. SOLUTION Locate the vertices, asmptote rectangle, and asmptotes in the original coordinate sstem [ig. 6(a)], then sketch the hperbola and translate the origin to the center of the hperbola [ig. 6(b)].

48 1008 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY 4 a b 3 3 (a) Asmptote rectangle (b) Hperbola Z igure 6 Net write the equation of the hperbola in the translated sstem: The origin in the translated sstem is at (h, k) (4, 3), and the translation formulas are h (4) 4 k 3 Thus, the equation of the hperbola in the original sstem is ( 3) ( 4) or, after simplifing and writing in the form of equation (1) on page 1001, MATCHED PROBLEM 4 ind the equation of an ellipse with foci on the line 4, minor ais on the line 3, length of the major ais 8, and length of the minor ais 4.

49 SECTION 11 4 Translation and Rotation of Aes 1009 ZZZ EXPLORE-DISCUSS Use the strateg of completing the square to transform each equation into an equation in an coordinate sstem. Note that the equation ou obtain is not one of the standard forms in Table 1; instead, it is either the equation of a degenerate conic or the equation has no solution. If the solution set of the equation is not empt, graph it and identif the graph (a point, a line, two parallel lines, or two intersecting lines). (A) (B) (C) 1 0 (D) (E) Z Rotation of Aes To handle the general equation of the form A B C D E 0 (1) when B 0, we need to be able to rotate, not just translate, coordinate aes. If ou hold a sheet of paper to a desk top with a pencil point, and move the paper without moving the pencil point, ou rotate the paper. Similarl, a rotation of coordinate aes occurs when the origin is kept fied and the and aes are obtained b rotating the and aes counterclockwise through an angle, as shown in igure 7. P (, ) (, ) r 0 0 Z igure 7

50 1010 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Referring to igure 7 and using trigonometr, we have r cos r sin (7) and r cos ( ) r sin ( ) (8) Using sum identities from trigonometr for the equations in (8), we obtain r cos ( ) r (cos cos sin sin ) r cos cos r sin sin (r cos ) cos (r sin ) sin cos sin Use sum identit for cosine. Distribute r. Use associative propert. Substitute r cos and r sin. (9) r sin ( ) r (sin cos cos sin ) r sin cos r cos sin (r cos ) sin (r sin ) cos sin cos Use sum identit for sine. Distribute r. Use associative propert. Substitute r cos and r sin. (10) Thus, equations (9) and (10) together transform the coordinate sstem into the coordinate sstem. Equations (9) and (10) can be solved for and in terms of and to produce formulas that transform the coordinate sstem back into the coordinate sstem. Omitting the details, the formulas for the transformation in the reverse direction are cos sin sin cos (11) These results are summarized in Theorem. Z THEOREM Rotation ormulas If the coordinate aes are rotated counterclockwise through an angle of, then the and coordinates of a point P are related b 1. cos sin. cos sin sin cos sin cos These formulas hold for P an point in the original coordinate sstem and an counterclockwise rotation.

51 SECTION 11 4 Translation and Rotation of Aes 1011 ZZZ EXPLORE-DISCUSS 3 Let be the first quadrant angle satisfing sin 3 and cos 4 and let an coordinate sstem be transformed into an coordinate sstem b a counterclockwise rotation through the angle. (A) Sketch the coordinate sstem in the coordinate sstem. (B) Epress and in terms of and. (C) Solve 0 to find the equation of the ais in the coordinate sstem. (D) Solve 0 to find the equation of the ais in the coordinate sstem. (E) Use the results found in parts C and D to graph the coordinate sstem in the coordinate sstem on a graphing calculator, using a squared viewing window. Z Rotation Used in Graphing We now investigate how rotation formulas are used in graphing. EXAMPLE Using the Rotation of Aes ormulas Transform the equation using a rotation of aes through 4. Graph the new equation and identif the curve. SOLUTION Use the rotation formulas: cos 4 sin 4 1 sin 4 cos 4 1 ( ) ( ) ( ) ( ) 1 ( ) 4 1 Substitute for and. Simplif. Distribute 1. Divide both sides b.

52 101 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY This is a standard equation for a hperbola. Summarizing, the graph of in the coordinate sstem is a hperbola with equation as shown in igure 8. Notice that the asmptotes in the rotated sstem are the and aes in the original sstem. Z igure 8 MATCHED PROBLEM Transform the equation 1 using a rotation of aes through 4. Graph the new equation and identif the curve. Check b graphing on a graphing calculator. In Eample, a 4 rotation transformed the original equation into one with no term. This made it eas to recognize that the graph of the transformed equation was a hperbola. In general, how do we determine the angle of rotation that will transform an equation with an term into one with no term? To find out, we substitute cos sin into equation (1) to obtain and sin cos A( cos sin ) B( cos sin )( sin cos ) C( sin cos ) D( cos sin ) E( sin cos ) 0 After multipling and collecting terms, we have where A B C D E 0 B (C A) sin cos B(cos sin ) (1) (13) or the term in equation (1) to drop out, B must be 0. We won t worr about A, C, D, and E at this point; the will automaticall be determined once we find so that B0. We set the right side of equation (13) equal to 0 and solve for : (C A) sin cos B(cos sin ) 0 Using the double-angle identities from trigonometr, sin sin cos and cos cos sin, we obtain (C A) sin B cos 0 B cos (A C) sin cos A C sin B cot A C B Add (A C) sin to both sides. Divide both sides b B sin. Use quotient identit. (14)

53 SECTION 11 4 Translation and Rotation of Aes 1013 Therefore, if we choose so that cot (A C)B, then B0 and the term in equation (1) will drop out. There is alwas an angle between 0 and 90 that solves equation (14), because the range of cot for 0 90 is the set of all real numbers (ig. 9) Z THEOREM 3 Angle of Rotation to Eliminate the Term Z igure 9 To transform the equation A B C D E 0 into an equation in and with no term, find so that cot A C B and and use the rotation formulas in Theorem. EXAMPLE 6 Identifing and Graphing an Equation with an Term Given the equation , find the angle of rotation so that the transformed equation will have no term. Sketch and identif the graph. SOLUTION (1) cot A C B Therefore, is a Quadrant II angle, and using the reference triangle in the figure, we can see that cos 4. We can find the rotation formulas eactl b the use of the half-angle identities sin B 1 cos and cos B 1 cos

54 1014 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Using these identities and substituting cos 4, we obtain and Hence, the rotation formulas (Theorem ) are and (16) Substituting equations (16) into equation (1), we have urther simplification leads to sin B 1 ( 4 ) cos B 1 ( 4 ) a b 6 a b a b 9 a b ( 3 ) 6 10 ( 3 )(3 ) 9 10 (3 ) 7 which is a standard equation for an ellipse. To graph, we rotate the original aes through an angle determined as follows: cot We could also use either sin or cos 1 110

55 SECTION 11 4 Translation and Rotation of Aes to determine the angle of rotation. Summarizing these results, the graph of in the coordinate sstem formed b a rotation of 71.7 is an ellipse with equation as shown in igure 10. Z igure 10 MATCHED PROBLEM 6 Given the equation , find the angle of rotation so that the transformed equation will have no term. Sketch and identif the graph. Check b graphing on a graphing calculator. Z Identifing Conics The discriminant of the general second-degree equation in two variables [equation (1)] is B 4AC. It can be shown that the value of this epression does not change when the aes are rotated. This forms the basis for Theorem 4. Z THEOREM 4 Identifing Conics The graph of the equation A B C D E 0 (1) is, ecluding degenerate cases, 1. A hperbola if B 4AC 0. A parabola if B 4AC 0 3. An ellipse if B 4AC 0 The proof of Theorem 4 is beond the scope of this book. Its use is best illustrated b eample. EXAMPLE 7 Identifing Conics Identif the following conics. (A) (B) (C) 4 4

56 1016 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY SOLUTIONS (A) The discriminant is B 4AC (1) 4(1)(1) so b Theorem 4 the conic is an ellipse. (B) The discriminant is B 4AC (1) 4(1)(1) 7 0 so b Theorem 4 the conic is a hperbola. (C) The discriminant is B 4AC (4) 4(1)(4) 0 so b Theorem 4 the conic is a parabola. MATCHED PROBLEM 7 Identif the following conics. (A) 10 (B) 10 (C) 10 Each of the equations in Eample 7 can be graphed b the method illustrated in Eample 6, or, as an alternative, b a graphing calculator. or eample, to graph the equation using a graphing calculator, first write the equation as a quadratic in the variable, then use the quadratic formula to solve for : 0 Write as a quadratic in. Use the quadratic formula with a 1, b, and c. () 4(1)( ) Simplif Graphing and 0 3 Z igure 11 produces the ellipse of Eample 7A (ig. 11).

57 SECTION 11 4 Translation and Rotation of Aes 1017 ANSWERS TO MATCHED PROBLEMS 1. 8; a parabola. (A) ( ) 4( 4); a parabola (B) 4 (C) (, 4) ( ) ( 1) 3. (A) 1; ellipse (B) 16 9 (C) (D) oci: (17, 1), (17, 1) ( 4) ( 3) 4. 1, or ; hperbola 6. 1; 30 ; hperbola (A) Ellipse (B) Hperbola (C) Parabola

58 1018 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY 11-4 Eercises In Problems 1 8: (A) ind translation formulas that translate the origin to the indicated point (h, k). (B) Write the equation of the curve for the translated sstem. (C) Identif the curve. 1. ( 3) ( ) 81; (3, ). ( 3) 8( ); (3, ) ( 7) ( 4) 3. 1; (7, 4) ( ) ( 6) 36; (, 6). ( 9) 16( 4); (4, 9) ( 9) ( ) 6. 1; (, 9) 10 6 ( 8) ( 3) 7. 1; (8, 3) 1 8 ( 7) ( 8) 8. 1; (7, 8) 0 In Problems 9 14: (A) Write each equation in one of the standard forms listed in Table 1. (B) Identif the curve ( 3) 9( ) ( ) 1( 3) ( ) ( 7) ( ) 8( 3) ( 6) 4( 4) ( 7) 7( 3) 8 In Problems 1 18, find the coordinates of the given points if the coordinate aes are rotated through the indicated angle. 1. (1, 0), (0, 1), (1, 1), (3, 4), (1, 0), (0, 1), (1, ), (, ), (1, 0), (0, 1), (1, ), (1, 3), (1, 1), (1, 1), (1, 1), (1, 1), 90 In Problems 19, find the equations of the and aes in terms of and if the coordinate aes are rotated through the indicated angle In Problems 3 30, transform each equation into one of the standard forms in Table 1. Identif the curve and graph it In Problems 31 36, find the coordinates of an foci relative to the original coordinate sstem. 31. Problem 3 3. Problem Problem 34. Problem 6 3. Problem Problem 30 In Problems 37 40, complete the square in each equation, identif the transformed equation, and graph If A 0, C 0, and E 0, find h and k so that the translation of aes h, k transforms the equation A C D E 0 into one of the standard forms of Table If A 0, C 0, and D 0, find h and k so that the translation of aes h, k transforms the equation A C D E 0 into one of the standard forms of Table 1.

59 SECTION 11 4 Translation and Rotation of Aes 1019 In Problems 43 46, find the transformed equation when the aes are rotated through the indicated angle. Sketch and identif the graph , 4 44., , , 4 In Problems 47, find the angle of rotation so that the transformed equation will have no term. Sketch and identif the graph In Problems 3 6, find the equations (in the original coordinate sstem) of the asmptotes of each hperbola. 3. ( 3) ( ) 1 4. ( 1) ( 4) ( 1) ( ) ( ) 16( ) ( 3) 9( 1) 9. 3( 4) ( ) In Problems 63 74, use the given information to find the equation of each conic. Epress the answer in the form A C D E 0 with integer coefficients and A A parabola with verte at (, ), ais the line, and passing through the point (, 1). 64. A parabola with verte at (4, 1), ais the line 1, and passing through the point (, 3). 6. An ellipse with major ais on the line 3, minor ais on the line, length of major ais 8, and length of minor ais An ellipse with major ais on the line 4, minor ais on the line 1, length of major ais 4, and length of minor ais. 67. An ellipse with vertices (4, 7) and (4, 3) and foci (4, 6) and (4, ). 68. An ellipse with vertices (3, 1) and (7, 1) and foci (1, 1) and (, 1). 69. A hperbola with transverse ais on the line, length of transverse ais 4, conjugate ais on the line 3, and length of conjugate ais. 70. A hperbola with transverse ais on the line, length of transverse ais 6, conjugate ais on the line, and length of conjugate ais An ellipse with the following graph: (3, 1) 7. An ellipse with the following graph: (, ) 73. A hperbola with the following graph: (, 4) (, ) (3, 1) (3, 3) (, 4) (0, ) (1, 1) (, ) (1, ) (4, 4)

60 100 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY 74. A hperbola with the following graph: (, 0) (3, 1) (3, 3) (, ) In Problems 81 and 8, use a rotation followed b a translation to transform each equation into a standard form. Sketch and identif the curve In Problems 7 80, use the discriminant to identif each graph. Graph on a graphing calculator Sstems of Nonlinear Equations Z Solving b Substitution Z Other Solution Methods If a sstem of equations contains an equations that are not linear, then the sstem is called a nonlinear sstem. In this section, we will investigate a special tpe of nonlinear sstem involving two first- or second-degree equations of the form A B C D E (1) Notice that the standard equations for a parabola, an ellipse, and a hperbola are seconddegree equations. It can be shown that a sstem of two equations of form (1) will have at most four solutions, some of which ma be imaginar. Since we are interested in finding both real and imaginar solutions to the sstems we consider, we now assume that the replacement set for each variable is the set of comple numbers, rather than the set of real numbers. Z Solving b Substitution The substitution method used to solve linear sstems of two equations in two variables is also an effective method for solving nonlinear sstems. This process is best illustrated b eamples.

61 SECTION 11 Sstems of Nonlinear Equations 101 EXAMPLE 1 Solving a Nonlinear Sstem b Substitution Solve the sstem: 3 1 SOLUTIONS Algebraic Solution We can start with either equation. But since the term in the second equation is a first-degree term with coefficient 1, our calculations will be simplified if we start b solving for in terms of in the second equation. Net we substitute for in the first equation to obtain an equation that involves alone: Graphical Solution We enter two equations to graph the circle and one to graph the line (ig. 1). Using the INTERSECT command, we find two solutions, (1, ) (ig. ) and (0.4,.) (, 11 ) (ig. 3) f Add 3 to both sides. Substitute for in the second equation. 6 6 S (1 3) ( 1)( ) 0 Square the binomial and collect like terms on the left side. Divide both sides b. actor. Use the zero product propert. Z igure 1 Z igure 4 4 1, If we substitute these values back into the equation 1 3, we obtain two solutions to the sstem: (1) 1 3( ) 11 Z igure 3 4 A check, which ou should provide, verifies that (1, ) and (, 11 are both solutions to the sstem. ) MATCHED PROBLEM 1 Solve the sstem: 10 1

62 10 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Refer to the algebraic solution of Eample 1. If we substitute the values of back into the equation, we obtain ( ) It appears that we have found two additional solutions, (1, ) and (, 11 ). But neither of these solutions satisfies the equation 3 1, which ou should verif. So, neither is a solution of the original sstem. We have produced two etraneous roots, apparent solutions that do not actuall satisf both equations in the sstem. This is a common occurrence when solving nonlinear sstems. It is alwas ver important to check the solutions of an nonlinear sstem to ensure that etraneous roots have not been introduced. ZZZ EXPLORE-DISCUSS 1 In Eample 1, we saw that the line 3 1 intersected the circle in two points. (A) Consider the sstem 3 10 Graph both equations in the same coordinate sstem. Are there an real solutions to this sstem? Are there an comple solutions? ind an real or comple solutions. (B) Consider the famil of lines given b 3 b b an real number What do all these lines have in common? Illustrate graphicall the lines in this famil that intersect the circle in eactl one point. How man such lines are there? What are the corresponding value(s) of b? What are the intersection points? How are these lines related to the circle? EXAMPLE Solving a Nonlinear Sstem b Substitution Solve:

63 SECTION 11 Sstems of Nonlinear Equations 103 SOLUTIONS Algebraic Solution Solve the second equation for, substitute into the first equation, and proceed as before. a b Divide both sides b. Substitute for in the first equation. Simplif. Graphical Solution Solving the first equation for, we have 0. 1 Subtract from both sides Multipl both sides b Take the square root of both sides u u 8 0 (u 4)(u ) 0 u 4, Thus, Multipl both sides b and simplif. Substitute u (see Section -6). actor. Use the zero product propert. We enter these two equations and / (ig. 4) in a graphing calculator. Using the INTERSECT command, we find the two real solutions, (, 1) (ig. ) and (, 1) (ig. 6). But we cannot find the two comple solutions. Z igure 4 4 or 1 i1 or, 1. or, 1. or i1, i1 i1. or i1, i1. i1 Z igure Thus, the four solutions to this sstem are (, 1), (, 1), (i1, i1), and (i1, i1). You should verif that each of these satisfies both equations in the sstem. 4 Z igure MATCHED PROBLEM Solve: 3 6 3

64 104 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY ZZZ EXPLORE-DISCUSS Stud the graphing calculator technique used in the graphical solution of Eample. Eplain wh this technique does not produce the imaginar solutions of a sstem of equations. EXAMPLE 3 Design An engineer is to design a rectangular computer screen with a 19-inch diagonal and a 17-square-inch area. ind the dimensions of the screen to the nearest tenth of an inch. SOLUTIONS Algebraic Solution Sketch a rectangle letting be the width and the height (ig. 7). We obtain the following sstem using the Pthagorean theorem and the formula for the area of a rectangle: Graphical Solution igure 8 shows the three functions required to graph this sstem. The graph is shown in igure 9. We are onl interested in the solutions in the first quadrant. Zooming in and using INTERSECT produces the results in igures 10 and 11. Assuming that the screen is wider than it is high, its dimensions are 1.0 inches b 11.7 inches. 19 inches Z igure 7 This sstem is solved using the procedures outlined in Eample. However, in this case, we are onl interested in real solutions. We start b solving the second equation for in terms of and substituting the result into the first equation. Z igure , ,6 0 Multipl both sides b and simplif. Subtract 361 from each side. Quadratic in. Z igure 9 40

65 SECTION 11 Sstems of Nonlinear Equations 10 Solve the last equation for using the quadratic formula, then solve for : 16 B (1)(30,6) inches or 11.7 inches Substitute each choice of into 17 to find the corresponding values: Z igure or 1.0 inches, inches 1 or 11.7 inches, inches Assuming the screen is wider than it is high, the dimensions are 1.0 inches b 11.7 inches. 10 Z igure 11 MATCHED PROBLEM 3 An engineer is to design a rectangular television screen with a 1-inch diagonal and a 09-square-inch area. ind the dimensions of the screen to the nearest tenth of an inch. Z Other Solution Methods We now look at some other techniques for solving nonlinear sstems of equations. EXAMPLE 4 Solving a Nonlinear Sstem b Elimination SOLUTIONS Solve: 17 Algebraic Solution This tpe of sstem can be solved using elimination b addition.* Multipl the second equation b 1 and add: Graphical Solution Solving each equation for gives us the four functions shown in igure 1. Eamining the graph in igure 13, we see that there are four intersection points. Using the INTERSECT command repeatedl (details omitted), we find that the solutions are (3, ), (3, ), (3, ), and (3, ). *This sstem can also be solved b substitution.

66 106 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Now substitute and back into either original equation to find. 4 or, or, 6 6 () () Thus, (3, ), (3, ), (3, ), and (3, ), are the four solutions to the sstem. The check of the solutions is left to ou. Z igure 1 Z igure 13 MATCHED PROBLEM 4 Solve: EXAMPLE Solving a Nonlinear Sstem Using actoring and Substitution Solve: SOLUTION 0 ( ) 0 0 or actor the left side of the equation that has a 0 constant term. Use the zero product propert. Thus, the original sstem is equivalent to the two sstems: These sstems are solved b substitution. or 3 0 IRST SYSTEM (0) (0) (1, 0) (1, 0) Substitute 0 in the second equation, and solve for. Simplif. Take the square root of both sides. Solutions to the first sstem.

67 SECTION 11 Sstems of Nonlinear Equations 107 SECOND SYSTEM Substitute in the second equation and solve for. Simplif. Divide both sides b. Take the square root of both sides. Substitute these values back into to find. (, ) (, ) Solutions to the second sstem. Combining the solutions for the first sstem with the solutions for the second sstem, the solutions for the original sstem are (1, 0), (1, 0), (, ), and (, ). The check of the solutions is left to ou. MATCHED PROBLEM Solve: 9 0 Eample is somewhat specialized. However, it suggests a procedure that is effective for some problems. EXAMPLE 6 Graphical Approimations of Real Solutions Use a graphing calculator to approimate real solutions to two decimal places: SOLUTION Before we can enter these equations in our calculator, we must solve for :

68 108 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY 4 1 Appling the quadratic formula to each equation, we have 6 4 ( 1) 0 ( 6) 0 a 1, b 4, c 1 a 1, b, c ( 1) ( 6) Since each equation has two solutions, we must enter four functions in the graphing calculator, as shown in igure 14(a). Eamining the graph in igure 14(b), we see that there are four intersection points. Using the INTERSECT command repeatedl (details omitted), we find that the solutions to two decimal places are (.10, 0.83), (0.37,.79), (0.37,.79), and (.10, 0.83) (a) (b) Z igure 14 MATCHED PROBLEM 6 Use a graphing calculator to approimate real solutions to two decimal places: ANSWERS TO MATCHED PROBLEMS 1. (1, 3), ( 9, 13 ). (13, 13), (13, 13), (i, 3i), (i, 3i) b 1. in. 4. (, 1), (, 1), (, 1), (, 1). (0, 3), (0, 3), (13, 13), (13, 13) 6. (3.89, 1.68), (0.96,.3), (0.96,.3), (3.89, 1.68)

69 SECTION 11 Sstems of Nonlinear Equations Eercises 1. Would ou choose substitution or elimination to solve the following nonlinear sstem? Assume a, b, c, d, e, and f 0. a b c d e f Justif our answer b describing the steps ou would take to solve this sstem.. Repeat Problem 1 for the following nonlinear sstem. a b c d e f Solve each sstem in Problems Solve each sstem in Problems An important tpe of calculus problem is to find the area between the graphs of two functions. To solve some of these problems it is necessar to find the coordinates of the points of intersections of the two graphs. In Problems 7 34, find the coordinates of the points of intersections of the two given equations. 7., 8., 3 9., 30., , 3. 3, , , Consider the circle with equation and the famil of lines given b b, where b is an real number. (A) Illustrate graphicall the lines in this famil that intersect the circle in eactl one point, and describe the relationship between the circle and these lines. (B) ind the values of b corresponding to the lines in part A, and find the intersection points of the lines and the circle. (C) How is the line with equation 0 related to this famil of lines? How could this line be used to find the intersection points in part B? 36. Consider the circle with equation and the famil of lines given b 3 4 b, where b is an real number. (A) Illustrate graphicall the lines in this famil that intersect the circle in eactl one point, and describe the relationship between the circle and these lines. (B) ind the values of b corresponding to the lines in part A, and find the intersection points of the lines and the circle. (C) How is the line with equation related to this famil of lines? How could this line be used to find the intersection points and the values of b in part B? Solve each sstem in Problems

70 1030 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY In Problems 4 0, use a graphing calculator to approimate the real solutions of each sstem to two decimal places enclosed b the fence (including the pool and the deck) is 1,1 square feet. ind the dimensions of the pool. ft Pool ft ence APPLICATIONS 1. NUMBERS ind two numbers such that their sum is 3 and their product is 1.. NUMBERS ind two numbers such that their difference is 1 and their product is 1. (Let be the larger number and the smaller number.) 3. GEOMETRY ind the lengths of the legs of a right triangle with an area of 30 square inches if its hpotenuse is 13 inches long. 4. GEOMETRY ind the dimensions of a rectangle with an area of 3 square meters if its perimeter is 36 meters long.. DESIGN An engineer is designing a small portable television set. According to the design specifications, the set must have a rectangular screen with a 7.-inch diagonal and an area of 7 square inches. ind the dimensions of the screen. 6. DESIGN An artist is designing a logo for a business in the shape of a circle with an inscribed rectangle. The diameter of the circle is 6. inches, and the area of the rectangle is 1 square inches. ind the dimensions of the rectangle. 6. inches 7. CONSTRUCTION A rectangular swimming pool with a deck feet wide is enclosed b a fence as shown in the figure. The surface area of the pool is 7 square feet, and the total area 8. CONSTRUCTION An open-topped rectangular bo is formed b cutting a 6-inch square from each corner of a rectangular piece of cardboard and bending up the ends and sides. The area of the cardboard before the corners are removed is 768 square inches, and the volume of the bo is 1,440 cubic inches. ind the dimensions of the original piece of cardboard. 6 in. 6 in. 6 in. 6 in. 6 in. 6 in. 6 in. 6 in. ft ft 9. TRANSPORTATION Two boats leave Bournemouth, England, at the same time and follow the same route on the 7-mile trip across the English Channel to Cherbourg, rance. The average speed of boat A is miles per hour greater than the average speed of boat B. Consequentl, boat A arrives at Cherbourg 30 minutes before boat B. ind the average speed of each boat. 60. TRANSPORTATION Bus A leaves Milwaukee at noon and travels west on Interstate 94. Bus B leaves Milwaukee 30 minutes later, travels the same route, and overtakes bus A at a point 10 miles west of Milwaukee. If the average speed of bus B is 10 miles per hour greater than the average speed of bus A, at what time did bus B overtake bus A?

71 Review 1031 CHAPTER 11 Review 11-1 Conic Sections; Parabola The plane curves obtained b intersecting a right circular cone with a plane are called conic sections. If the plane cuts clear through one nappe, then the intersection curve is called a circle if the plane is perpendicular to the ais and an ellipse if the plane is not perpendicular to the ais. If a plane cuts onl one nappe, but does not cut clear through, then the intersection curve is called a parabola. If a plane cuts through both nappes, but not through the verte, the resulting intersection curve is called a hperbola. A plane passing through the verte of the cone produces a degenerate conic a point, a line, or a pair of lines. The figure illustrates the four nondegenerate conics. directri is called the ais of smmetr, and the point on the ais halfwa between the directri and focus is called the verte. L V(Verte) P d 1 d d 1 d (ocus) Directri Ais of smmetr Parabola rom the definition of a parabola, we can obtain the following standard equations: Standard Equations of a Parabola with Verte at (0, 0) 1. 4a Verte (0, 0) ocus: (a, 0) Directri: a Smmetric with respect to the ais Ais of smmetr the ais Circle Ellipse 0 0 a 0 (opens left) a 0 (opens right) The graph of Parabola Hperbola A B C D E 0 is a conic, a degenerate conic, or the empt set. The following is a coordinate-free definition of a parabola: Parabola A parabola is the set of all points in a plane equidistant from a fied point and a fied line L (not containing ) in the plane. The fied point is called the focus, and the fied line L is called the directri. A line through the focus perpendicular to the. 4a Verte: (0, 0) ocus: (0, a) Directri: a Smmetric with respect to the ais Ais of smmetr the ais 0 a 0 (opens down) 0 a 0 (opens up)

72 103 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY 11- Ellipse The following is a coordinate-free definition of an ellipse: Ellipse An ellipse is the set of all points P in a plane such that the sum of the distances from P to two fied points in the plane is a constant. (The constant is required to be greater than the distance between the two fied points.) Each of the fied points, and, is called a focus, and together the are called foci. Referring to the figure, the line segment VV through the foci is the major ais. The perpendicular bisector BB of the major ais is the minor ais. Each end of the major ais, V and V, is called a verte. The midpoint of the line segment is called the center of the ellipse.. a 7 b 7 0 b a 1 intercepts: b intercepts: a (vertices) oci: (0, c), (0, c) c a b Major ais length a Minor ais length b a c a rom the definition of an ellipse, we can obtain the following standard equations: Standard Equations of an Ellipse with Center at (0, 0) V 1. a 7 b 7 0 a b 1 intercepts: a (vertices) intercepts: b oci: (c, 0), (c, 0) c a b Major ais length a Minor ais length b a d 1 d Constant B d 1 b c 0 c a B a P d V [Note: Both graphs are smmetric with respect to the ais, ais, and origin. Also, the major ais is alwas longer than the minor ais.] 11-3 Hperbola The following is a coordinate-free definition of a hperbola: Hperbola b c a A hperbola is the set of all points P in a plane such that the absolute value of the difference of the distances from P to two fied points in the plane is a positive constant. (The constant is required to be less than the distance between the two fied points.) Each of the fied points, and, is called a focus. The intersection points V and V of the line through the foci and the two branches of the hperbola are called vertices, and each is called a verte. The line segment VV is called the transverse ais. The midpoint of the transverse ais is the center of the hperbola. d 1 d Constant V 0 b P d 1 d V b

73 Review 1033 rom the definition of a hperbola, we can obtain the following standard equations: Standard Equations of a Hperbola with Center at (0, 0) 1. a b 1 intercepts: a (vertices) intercepts: none oci: (c, 0), (c, 0) c a b Transverse ais length a Conjugate ais length b Asmptotes: b a 11-4 Translation and Rotation of Aes In Sections 11-1, 11-, and 11-3 we found standard equations for parabolas, ellipses, and hperbolas located with their aes on the coordinate aes and centered relative to the origin. We now move the conics awa from the origin while keeping their aes parallel to the coordinate aes. In this process we obtain new standard equations that are special cases of the equation A C D E 0, where A and C are not both zero. The basic mathematical tool used is translation of aes. A translation of coordinate aes occurs when the new coordinate aes have the same direction as, and are parallel to, the original coordinate aes. Translation formulas are as follows: 1. h. h k k where (h, k) are the coordinates of the origin 0 relative to the original sstem. c a b b c a c P (, ) (, ). a b 1 intercepts: none intercepts: a (vertices) oci: (0, c), (0, c) c a b Transverse ais length a Conjugate ais length b Asmptotes: a b b c a a c [Note: Both graphs are smmetric with respect to the ais, ais, and origin.] c b Table 1 on page 1034 lists the standard equations for conics. If the coordinate aes are rotated counterclockwise through an angle into the coordinate aes, then the and coordinate sstems are related b the rotation formulas: 1. cos sin. cos sin sin cos To transform the general quadratic equation sin cos A B C D E 0 into an equation in and with no term, choose the angle of rotation to satisf cot (A C)B and The discriminant of the general second-degree equation in two variables is B 4AC and the graph is 1. A hperbola if B 4AC 0. A parabola if B 4AC 0 3. An ellipse if B 4AC 0 0 (0, 0) (0, 0) (h, k) 0

74 1034 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Table 1 Standard Equations for Conics Parabolas ( h) 4a( k) ( k) 4a( h) Verte (h, k) ocus (h, k a) a 0 opens up a 0 opens down a V (h, k) Verte (h, k) ocus (h a, k) a a 0 opens left a 0 opens right V (h, k) Circles ( h) ( k) r Center (h, k) Radius r r C (h, k) ( h) a ( k) 1 b Ellipses a b 0 ( h) b ( k) 1 a b a (h, k) Center (h, k) Major ais a Minor ais b a Center (h, k) Major ais a Minor ais b (h, k) b Hperbolas (h, k) ( h) b a a ( k) b 1 Center (h, k) Transverse ais a Conjugate ais b (h, k) ( k) a a b ( h) 1 b Center (h, k) Transverse ais a Conjugate ais b

75 Review Eercises Sstems of Nonlinear Equations If a sstem of equations contains an equations that are not linear, then the sstem is called a nonlinear sstem. In this section we investigated nonlinear sstems involving second-degree terms such as It can be shown that such sstems have at most four solutions, some of which ma be imaginar. Several methods were used to solve nonlinear sstems of the indicated form: solution b substitution, solution using elimination b addition, and solution using factoring and substitution. It is alwas important to check the solutions of an nonlinear sstem to ensure that etraneous roots have not been introduced. CHAPTER 11 Review Eercises Work through all the problems in this chapter review and check answers in the back of the book. Answers to all review problems are there, and following each answer is a number in italics indicating the section in which that tpe of problem is discussed. Where weaknesses show up, review appropriate sections in the tet. In Problems 1 6, graph each equation and locate foci. Locate the directri for an parabolas. ind the lengths of major, minor, transverse, and conjugate aes where applicable In Problems 7 9: (A) Write each equation in one of the standard forms listed in Table 1 of the review. (B) Identif the curve. 7. 4( ) ( 4) ( ) 1( 4) ( 6) 9( 4) ind the coordinates of the point (3, 4) when the aes are rotated through (A) 30 (B) 4 (C) ind the equations of the and aes in terms of and if the aes are rotated through an angle of 7. In Problems 1 14, solve the sstem ind the equation of the parabola having its verte at the origin, its ais of smmetr the ais, and (4, ) on its graph. In Problems 16 and 17, find the equation of the ellipse in the form M N 1 if the center is at the origin, and: 16. Major ais on ais 17. Major ais on ais Major ais length 1 Minor ais length 1 Minor ais length 10 Distance between foci 16 In Problems 18 and 19, find the equation of the hperbola in the form or M N 1 M if the center is at the origin, and: 18. Transverse ais on ais Conjugate ais length 6 Distance between foci Transverse ais on ais Transverse ais length 14 Conjugate ais length 16 In Problems 0, solve the sstem. M, N N 1 M, N ind the equation of the parabola having directri and focus (0, ). 7. ind the foci of the ellipse through the point (6, 0) if the center is at the origin, the major ais is on the ais, and the major ais has twice the length of the minor ais.

76 1036 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY 8. ind the intercepts of a hperbola if the center is at the origin, the conjugate ais is on the ais and has length 4, and (0, 3) is a focus. 9. ind the directri of a parabola having its verte at the origin and focus (4, 0). 30. ind the points of intersection of the parabolas 8 and. 31. ind the intercepts of an ellipse if the center is at the origin, the major ais is on the ais and has length 14, and (0, 1) is a focus. 3. ind the foci of the hperbola through the point (0, 4) if the center is at the origin, the transverse ais is on the ais, and the conjugate ais has twice the length of the transverse ais. In Problems 33 3, transform each equation into one of the standard forms in Table 1 in the review. Identif the curve and graph it Given the equation , find the transformed equation when the aes are rotated through 30. Sketch and identif the graph. 37. Given the equation 6 7 0, find the angle of rotation so that the transformed equation will have no term. Sketch and identif the graph. 38. Given the equation , identif the curve. 39. Use the definition of a parabola and the distance formula to find the equation of a parabola with directri 6 and focus at (, 4). 40. ind an equation of the set of points in a plane each of whose distance from (4, 0) is twice its distance from the line 1. Identif the geometric figure. 41. ind an equation of the set of points in a plane each of whose distance from (4, 0) is two-thirds its distance from the line 9. Identif the geometric figure. In Problems 4 44, find the coordinates of an foci relative to the original coordinate sstem. 4. Problem Problem Problem 3 In Problems 4 47, find the equations of the asmptotes of each hperbola APPLICATIONS 48. COMMUNICATIONS A parabolic satellite television antenna has a diameter of 8 feet and is 1 foot deep. How far is the focus from the verte? 49. ENGINEERING An elliptical gear is to have foci 8 centimeters apart and a major ais 10 centimeters long. Letting the ais lie along the major ais (right positive) and the ais lie along the minor ais (up positive), write the equation of the ellipse in the standard form a b 1 0. SPACE SCIENCE A hperbolic reflector for a radio telescope (such as that illustrated in Problem 9, Eercises 11-3) has the equation If the reflector has a diameter of 30 feet, how deep is it? Compute the answer to three significant digits. CHAPTER 11 ZZZ GROUP ACTIVITY ocal Chords Man of the applications of the conic sections are based on their reflective or focal properties. One of the interesting algebraic properties of the conic sections concerns their focal chords.

77 Cumulative Review 1037 If a line through a focus contains two points G and H of a conic section, then the line segment GH is called a focal chord. Let G ( 1, 1 ) and H (, ) be points on the graph of 4a such that GH is a focal chord. Let u denote the length of G and v the length of H (ig. 1). G u v H (a, a) Z igure 1 ocal chord GH of the parabola 4a. (A) Use the distance formula to show that u 1 a. (B) Show that G and H lie on the line a m, where m ( 1 )( 1 ). (C) Solve a m for and substitute in 4a, obtaining a quadratic equation in. Eplain wh 1 a. 1 (D) Show that u 1 v 1 a. (u a) (E) Show that u v 4a Eplain wh this implies that u v 4a, with equalit if and onl if u a. u v a. () Which focal chord is the shortest? Is there a longest focal chord? 1 (G) Is a constant for focal chords of the ellipse? or focal chords of the hperbola? Obtain evidence for u 1 v our answers b considering specific eamples. CHAPTERS Cumulative Review Work through all the problems in this cumulative review and check answers in the back of the book. Answers to all review problems are there, and following each answer is a number in italics indicating the section in which that tpe of problem is discussed. Where weaknesses show up, review appropriate sections in the tet. 1. Determine whether each of the following can be the first three terms of an arithmetic sequence, a geometric sequence, or neither. (A) 0, 1, 10,... (B),, 1,... (C),, 0,... (D) 7, 9, 3,... (E) 9, 6, 3,... In Problems 4: (A) Write the first four terms of each sequence. (B) ind a 8. (C) ind S 8.. a n n 3. a n 3n 1 4. a 1 100; a n a n1 6, n. Evaluate each of the following: 3! 9! (A) 8! (B) (C) 30! 3!(9 3)!

78 1038 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY 6. Evaluate each of the following: (A) a 7 (B) (C) b In Problems 7 9, graph each equation and locate foci. Locate the directri for an parabolas. ind the lengths of major, minor, transverse, and conjugate aes where applicable Solve What tpe of curve is the graph of 1. A coin is flipped three times. How man combined outcomes are possible? Solve (A) B using a tree diagram (B) B using the multiplication principle 13. How man was can four distinct books be arranged on a shelf? Solve (A) B using the multiplication principle (B) B using permutations or combinations, whichever is applicable 14. In a single deal of 3 cards from a standard -card deck, what is the probabilit of being dealt three diamonds? 1. Each of the 10 digits 0 through 9 is printed on 1 of 10 different cards. our of these cards are drawn in succession without replacement. What is the probabilit of drawing the digits 4,, 6, and 7 b drawing 4 on the first draw, on the second draw, 6 on the third draw, and 7 on the fourth draw? What is the probabilit of drawing the digits 4,, 6, and 7 in an order? 16. A thumbtack lands point down in 38 out of 100 tosses. What is the approimate empirical probabilit of the tack landing point up? Verif Problems 17 and 18 for n 1,, and P n : (4n 3) n(n 1) 18. P n : n n is divisible b In Problems 19 and 0, write and 19. or in Problem or in Problem 18 P n C 7, P k P 7, P k1. 1. ind the equation of the parabola having its verte at the origin, its ais the ais, and (, 8) on its graph. P n. ind an equation of an ellipse in the form if the center is at the origin, the major ais is the ais, the major ais length is 10, and the distance of the foci from the center is ind an equation of a hperbola in the form if the center is at the origin, the transverse ais length is 16, and the distance of the foci from the center is 189. In Problems 4 and, find the angle of rotation so that the transformed equation will have no term. Identif the curve and graph it In Problems 6 and 7, solve the sstem ind all real solutions to two decimal places k k k1 9. Write a without summation notation and find the sum Write the series using! 3! 4!! 6! 7! summation notation with the summation inde k starting at k ind for the geometric series S M N 1 M N M, N 7 0 M, N How man four-letter code words are possible using the first si letters of the alphabet if no letter can be repeated? If letters can be repeated? If adjacent letters cannot be alike? 33. A basketball team with 1 members has two centers. If plaers are selected at random, what is the probabilit that both centers are selected? Epress the answer in terms of C n,r or P n,r, as appropriate, and evaluate. 34. A single die is rolled 1,000 times with the frequencies of outcomes shown in the table. (A) What is the approimate empirical probabilit that the number of dots showing is divisible b 3?

79 Cumulative Review 1039 (B) What is the theoretical probabilit that the number of dots showing is divisible b 3? Number of dots facing up requenc Let a n 100(0.9) n and b n n. ind the least positive integer n such that a n 6 b n b graphing the sequences { a n } and { b n } with a graphing calculator. Check our answer b using a graphing calculator to displa both sequences in table form. 36. Evaluate each of the following: (A) (B) C(, ) (C) 37. Epand (a 1 b) 6 using the binomial formula. 38. ind the fifth and the eighth terms in the epansion of (3 ) 10. Establish each statement in Problems 39 and 40 for all positive integers using mathematical induction. 39. in Problem in Problem ind the sum of all the odd integers between 0 and Use the formula for the sum of an infinite geometric series to write as the quotient of two integers. 43. Let a k a 30 for k 0, 1,..., 30. Use a k b(0.1)30k (0.9) k graphing calculator to find the largest term of the sequence and the number of terms that are greater than In Problems 44 46, use a translation of coordinates to transform each equation into a standard equation for a nondegenerate conic. Identif the curve and graph it How man nine-digit zip codes are possible? How man of these have no repeated digits? 48. Use mathematical induction to prove that the following statement holds for all positive integers: P n : P n P, P n a 0 b a k 6 1 (n 1)(n 1) n n Three-digit numbers are randoml formed from the digits 1,, 3, 4, and. What is the probabilit of forming an even number if digits cannot be repeated? If digits can be repeated? 0. Use the binomial formula to epand ( i) 6, where i is the imaginar unit. 1. Use the definition of a parabola and the distance formula to find the equation of a parabola with directri 3 and focus (6, 1).. An ellipse has vertices (4, 0) and foci (, 0). ind the intercepts. 3. A hperbola has vertices (, 3) and foci (, ). ind the length of the conjugate ais. 4. Seven distinct points are selected on the circumference of a circle. How man triangles can be formed using these seven points as vertices?. Use mathematical induction to prove that n 6 n! for all integers n Use mathematical induction to show that a n 6 b n 6, where a 1 3, a n a n1 1 for n 7 1, and b n n 1, n ind an equation of the set of points in the plane each of whose distance from (1, 4) is three times its distance from the ais. Write the equation in the form A C D E 0, and identif the curve. 8. A bo of 1 lightbulbs contains 4 defective bulbs. If three bulbs are selected at random, what is the probabilit of selecting at least one defective bulb? APPLICATIONS 9. ECONOMICS The government, through a subsid program, distributes $,000,000. If we assume that each individual or agenc spends 7% of what it receives, and 7% of this is spent, and so on, how much total increase in spending results from this government action? 60. GEOMETRY ind the dimensions of a rectangle with perimeter 4 meters and area 3 square meters. 61. ENGINEERING An automobile headlight contains a parabolic reflector with a diameter of 8 inches. If the light source is located at the focus, which is 1 inch from the verte, how deep is the reflector? 6. ARCHITECTURE A sound whispered at one focus of a whispering chamber can be easil heard at the other focus. Suppose that a cross section of this chamber is a semielliptical arch

80 1040 CHAPTER 11 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY which is 80 feet wide and 4 feet high (see the figure). How far is each focus from the center of the arch? How high is the arch above each focus? Part affiliation Age Democrat Republican Independent Totals Under feet Over Totals , feet 63. POLITICAL SCIENCE A random surve of 1,000 residents in a state produced the following results: ind the empirical probabilit that a person selected at random: (A) Is under 30 and a Democrat (B) Is under 40 and a Republican (C) Is over 9 or is an Independent