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