SECTION 9-1 Conic Sections; Parabola

Transcription

1 66 9 Additional Topics in Analtic Geometr Analtic geometr, a union of geometr and algebra, enables us to analze certain geometric concepts algebraicall and to interpret certain algebraic relationships geometricall. Our two main concerns center around graphing algebraic equations and finding equations of useful geometric figures. We have discussed a number of topics in analtic geometr, such as straight lines and circles, in earlier chapters. In this chapter we discuss additional analtic geometr topics: conic sections and translation of aes. René Descartes ( ), the rench philosopher mathematician, is generall recognized as the founder of analtic geometr. SECTION 9-1 Conic Sections; Conic Sections Definition of a Drawing a Standard Equations and Their Graphs 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. Conic Sections In Section 3-2 we found that the graph of a first-degree equation in two variables, A B C (1) where A and B are not both, 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 second-degree equation in two variables, A 2 B C 2 D E (2) V L Constant where A, B, and C are not all, ield for different sets of values of the coefficients? The graphs of equation (2) 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. 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, Nappe *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.

2 9-1 Conic Sections; 67 IGURE 1 Conic sections. Circle Ellipse Hperbola 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 (see ig. 2). We will discuss man applications of conics throughout the remainder of this chapter. IGURE 2 Eamples of conics. Water from Lamp light garden hose Serving platter shadow Wheel (circle) (parabola) (ellipse) (hperbola) (a) (b) (c) (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 Section 3-1 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. Definition of a The following definition of a parabola does not depend on the coordinates of points in an coordinate sstem:

3 68 9 Additional Topics in Analtic Geometr DEINITION 1 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 2 d 1 d 2 Ais V(Verte) (ocus) Directri Drawing a Using the definition, 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. Since 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. IGURE 3 Drawing a parabola. String C D A E B EXPLORE-DISCUSS 1 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. Standard Equations and Their Graphs Using the definition of a parabola and the distance-between-two-points formula d ( 2 1 ) 2 ( 2 1 ) 2 (3)

4 9-1 Conic Sections; 69 IGURE 4 with center at the origin and ais the ais. d 1 M( a, ) P(, ) P(, ) d 1 M( a, ) a Directri a d 2 ocus (a, ) d 2 ocus (a, ) a Directri a a, focus on positive ais (a) a, focus on negative ais (b) 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, ). 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 and to the left if a. The verte is at the origin, the directri is a, and the coordinates of M are ( a, ). The point P(, ) is a point on the parabola if and onl if d 1 d 2 d(p, M) d(p, ) ( a) 2 ( ) 2 ( a) 2 ( ) 2 ( a) 2 ( a) a a 2 2 2a a a Use equation (3). Square both sides. Simplif. (4) Equation (4) is the standard equation of a parabola with verte at the origin, ais the ais, and focus at (a, ). Now we locate the verte at the origin and focus on the ais at (, a). Looking at igure 5 on the following page, we note that the parabola opens upward if a and downward if a. The directri is a, and the coordinates of N are (, a). The point P(, ) is a point on the parabola if and onl if d 1 d 2 d(p, N) d(p, ) ( ) 2 ( a) 2 ( ) 2 ( a) 2 ( a) 2 2 ( a) 2 2 2a a a a 2 2 4a Use equation (3). Square both sides. Simplif. (5) Equation (5) is the standard equation of a parabola with verte at the origin, ais the ais, and focus at (, a).

5 61 9 Additional Topics in Analtic Geometr IGURE 5 with center at the origin and ais the ais. Directri a (, a) ocus a d 2 P(, ) d 1 N(, a) N(, a) d 1 P(, ) a d2 (, a) ocus Directri a a, focus on positive ais (a) a, focus on negative ais (b) We summarize these results for eas reference in Theorem 1: Theorem 1 Standard Equations of a with Verte at (, ) a Verte: (, ) ocus: (a, ) Directri: a Smmetric with respect to the ais Ais the ais a (opens left) a (opens right) a Verte: (, ) ocus: (, a) Directri: a Smmetric with respect to the ais Ais the ais a (opens down) a (opens up) EXAMPLE 1 Graphing 2 4a Graph 2 16, and locate the focus and directri. Solution To graph 2 16, it is convenient to assign values that make the right side a perfect square, and solve for. Note that must be or negative for to be real. Since the coefficient of is negative, a must be negative, and the parabola opens downward (ig. 6).

6 9-1 Conic Sections; Directri 4 a ocus: ( 4) (, a) (, 4) Directri: a ( 4) (, 4) 1 IGURE Matched Problem 1 Graph 2 8, and locate the focus and directri. Remark. To graph the equation 2 16 of Eample 1 on a graphing utilit, we first solve the equation for and then graph the function If that same approach is used to graph the equation 2 8 of Matched Problem 1, then 8, and there are two functions to graph. The graph of 8 is the upper half of the parabola, and the graph of 8 is the lower half (see ig. 7). IGURE CAUTION A common error in making a quick sketch of 2 4a or 2 4a is to sketch the first with the ais as its ais and the second with the ais as its ais. The graph of 2 4a is smmetric with respect to the ais, and the graph of 2 4a is smmetric with respect to the ais, as a quick smmetr check will reveal. EXAMPLE 2 inding the Equation of a (A) ind the equation of a parabola having the origin as its verte, the ais as its ais, and ( 1, 5) on its graph. (B) ind the coordinates of its focus and the equation of its directri.

7 612 9 Additional Topics in Analtic Geometr Solutions (A) The parabola is opening down and has an equation of the form 2 4a. Since ( 1, 5) is on the graph, we have Thus, the equation of the parabola is 2 4a ( 1) 2 4a( 5) 1 2a a 5 2 4( 5) 2 a (B) ocus: 2 2 4( 5) (, a) (, 5) Directri: a ( 5) 5 Matched Problem 2 (A) ind the equation of a parabola having the origin as its verte, the ais as its ais, and (4, 8) on its graph. (B) ind the coordinates of its focus and the equation of its directri. EXPLORE-DISCUSS 2 Consider the graph of an equation in the variables and. The equation of its magnification b a factor k is obtained b replacing and in the equation b /k and /k, respectivel. (Of course, a magnification b a factor k between and 1 means an actual reduction in size.) (A) Show that the magnification b a factor 3 of the circle with equation has equation (B) Eplain wh ever circle with center at (, ) is a magnification of the circle with equation (C) ind the equation of the magnification b a factor 3 of the parabola with equation 2. Graph both equations. (D) Eplain wh ever parabola with verte (, ) that opens upward is a magnification of the parabola with equation 2.

8 9-1 Conic Sections; 613 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 utilize parabolic forms in their design. igure 8(a) illustrates a parabolic reflector used in all reflecting telescopes from 3- to 6-inch home tpe to the 2-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, C have been achieved b such furnaces. If we locate a light source at, then the ras in igure 8(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 8(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. igure 8(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. IGURE 8 Uses of parabolic forms. Parallel light ras Parabolic reflector Suspension bridge Arch bridge (a) (b) (c) EXAMPLE 3 Parabolic Reflector A paraboloid is formed b revolving a parabola about its ais. A spotlight in the form of a paraboloid 5 inches deep has its focus 2 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 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. Since we are in charge, we can make things simpler for ourselves b locating the verte at the origin and choosing a coordinate ais as the ais. We choose the ais as the ais of the parabola with the parabola opening upward. See igure 9 on the following page.

9 614 9 Additional Topics in Analtic Geometr 5 R (R, 5) Step 2. ind the equation of the parabola in the figure. Since the parabola has the ais as its ais and the verte at the origin, the equation is of the form (,2) Spotlight 2 4a 5 IGURE 9 5 We are given (, a) (, 2); thus, a 2, and the equation of the parabola is 2 8 Step 3. Use the equation found in step 2 to find the radius R of the opening. Since (R, 5) is on the parabola, we have R 2 8(5) R inches Matched Problem 3 Repeat Eample 3 with a paraboloid 12 inches deep and a focus 9 inches from the verte. Answers to Matched Problems 1. ocus: ( 2, ) Directri: ( 2, ) Directri (A) 2 16 (B) ocus: (4, ); Directri: 4 3. R 2.8 in. EXERCISE 9-1 A In Problems 1 12, graph each equation, and locate the focus and directri ind the coordinates to two decimal places of the focus for each parabola in Problems B In Problems 19 24, find the equation of a parabola with verte at the origin, ais the or ais, and:

10 9-1 Conic Sections; ocus ( 6, ) 2. Directri Directri ocus (3, ) ocus (, 1 ) 24. Directri 3 2 In Problems 25 3, find the equation of the parabola having its verte at the origin, its ais as indicated, and passing through the indicated point. 25. ais; ( 4, 2) 26. ais; (3, 15) 27. ais; (9, 27) 28. ais; (121, 11) 29. ais; ( 8, 2) 3. ais; ( 2, 3) In Problems 31 34, find the first-quadrant points of intersection for each sstem of equations to three decimal places. C In Problems 43 46, use the definition of a parabola and the distance formula to find the equation of a parabola with: 43. Directri 4 and focus (2, 2) 44. Directri 2 and focus ( 3, 6) 45. Directri 2 and focus (6, 4) 46. Directri 3 and focus (1, 4) In Problems 47 5, use a graphing utilit to find the coordinates of all points of intersection to two decimal places , , , , Check Problems with a graphing utilit Consider the parabola with equation 2 4a. (A) How man lines through (, ) 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 (, ) having slope m. APPLICATIONS 51. Engineering. The parabolic arch in the concrete bridge in the figure must have a clearance of 5 feet above the water and span a distance of 2 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 the parabola. 36. ind the coordinates of all points of intersection of the parabola with equation 2 4a and the parabola with equation 2 4b. 37. If a line through the focus contains two points A and B of a parabola, then the line segment AB is called a focal chord. ind the coordinates of A and B for the focal chord that is perpendicular to the ais of the parabola 2 4a. A 2 4a (, a) B 52. Astronom. The cross section of a parabolic reflector with 6-inch diameter is ground so that its verte is.15 inch below the rim (see the figure). igure for 37 and 38 6 inches.15 inch 38. ind the length of the focal chord AB that is perpendicular to the ais of the parabola 2 4a. In Problems 39 42, determine whether the statement is true or false. If true, eplain wh. If false, give a countereample. 39. If a is real, then the graph of 2 4a is a parabola. 4. If a is negative, then the graph of 2 4a is a parabola. 41. Ever vertical line intersects the graph of Ever nonhorizontal line intersects the graph of 2 4. Parabolic reflector (A) ind the equation of the parabola after inserting an coordinate sstem with the verte at the origin, the ais (pointing upward) the ais of the parabola. (B) How far is the focus from the verte?

11 616 9 Additional Topics in Analtic Geometr 53. Space Science. A designer of a 2-foot-diameter parabolic electromagnetic antenna for tracking space probes wants to place the focus 1 feet above the verte (see the figure). (A) ind the equation of the parabola using the ais of the parabola as the ais (up positive) and verte at the origin. (B) Determine the depth of the parabolic reflector. 54. Signal Light. A signal light on a ship is a spotlight with parallel reflected light ras (see the figure). Suppose the parabolic reflector is 12 inches in diameter and the light source is located at the focus, which is 1.5 inches from the verte. 2 ft Signal light ocus ocus Radiotelescope 1 ft (A) ind the equation of the parabola using the ais of the parabola as the ais (right positive) and verte at the origin. (B) Determine the depth of the parabolic reflector. SECTION 9-2 Ellipse Definition of an Ellipse Drawing an Ellipse Standard Equations and Their Graphs 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. Definition of an Ellipse The following is a coordinate-free definition of an ellipse: DEINITION 1 Ellipse An ellipse is the set of all points P in a plane such that the sum of the distances of P from two fied points in the plane is constant. 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,