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-1 535 52. Signal Light. A signal light on a ship is a spotlight with parallel reflected light rays (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 vertex. (A) Find the equation of the parabola using the axis of the parabola as the x axis (right positive) and vertex at the origin. (B) Determine the depth of the parabolic reflector. Section 7-2 Ellipse Definition of an Ellipse Drawing an Ellipse Standard Equations and Their Graphs Applications 7-2 Ellipse Signal light Focus DEFINITION 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 equa- tions for ellipses specially located in a rectangular coordinate system. Definition of an 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 of P from two fixed points in the plane is constant. Each of the fixed points, F' and F, is called a focus, and together they are called foci. Referring to the figure, the line segment V' V through the foci is the major axis. The perpendicular bisector B' B of the major axis is the minor axis. Each end of the major axis, V' and V, is called a vertex. The midpoint of the line segment F'F is called the center of the ellipse. dl + d2 = Constant dl P
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536 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Draunng an Ellipse An ellipse is easy to draw. All you need is a piece of string, two thumbtacks, and a pencil or pen (see Fig. 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. Finally, catch the tip of a pencil under the string and move it while keeping the string taut. The resulting figure is by definition an ellipse. Ellipses of different shapes result, depending on the placement of thumb- tacks and the length of the string joining them. FIGURE I Drawing an ellipse. String Focus Note that dl + d2 always adds up to the length of the string, which does not change. Focus Standard Equations 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 coor- dinate system. We start by placing an ellipse in the coordinate system with the foci on the x axis equidistant from the origin at F'(—c, 0) and F(c, 0), as in Figure 2. FIGURE 2 Ellipse with foci on x axis. x dl + d2 = Constant For reasons that will become clear soon, it is convenient to represent the con- stant sum dl + d2 by 2a, a > 0. Also, the geometric fact that the sum of the lengths of any two sides of a triangle must be greater than the third side can be applied to Figure 2 to derive the following useful result: d(F', P) + d(P, F) > d(F', F) -k 612 (1)
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-1 537 7-2 Ellipse We will use this result in the derivation of the equation of an ellipse, which we now begin. Referring to Figure 2, the point P(x, y) is on the ellipse if and only if — 2a — 2a b2 _ b2 _ Cll + — d(P, F') + d(P, F) 2a After eliminating radicals and simplifying, a good exercise for you, we obtain (a2 — c2)x2 + a2y2 — a2(a2 — c2) (2) (3) 2 a 2 a Dividing both sides of equation (2) by a2(a2 — c2) is permitted, since neither a 2 2 — c2 is 0. From equation (1), a > c; thus 612 > c2 and a 2 - c2 > O. The nor a constant a was chosen positive at the beginning. To simplify equation (3) further, we let 2 a to obtain 2 x 2 + 12 = 1 a 2 (4) (5) From equation (5) we see that the x intercepts are x — ±a and the y intercepts ±b. The x intercepts are also the vertices. Thus, are y Major axis length = 2a Minor axis length = 2b To see that the major axis is longer than the minor axis, we show that 2a > 2b. Returning to equation (4), (b 2 a 2 b2 < 612 b2 — a2 < 0 — a)(b + a) < 0 2b < 261 261 > 2b 2 Definition of < Since b + a is positive, b negative. — a must be Length of Length of major axis minor axis
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538 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY If we start with the foci on the y axis at F(O, c) and F' (0, —c) as in Figure 3, instead of on the x axis as in Figure 2, then, following arguments similar to those used for the first derivation, we obtain b2 _ —c) 2d — 2a a where the relationship among a, b, and c remains the same as before: 2 a 2 (6) (7) The center is still at the origin, but the major axis is now along the y axis and the minor axis is along the x axis. FIGURE 3 Ellipse with foci on y axis. dl x dl + d2 Constant To sketch graphs of equations of the form of equations (5) or (6) is an easy matter. We find the x and y intercepts and sketch in an appropriate ellipse. Since replacing x with —x or y with —y produces an equivalent equation, we conclude that the graphs are symmetric with respect to the x axis, y axis, and origin. If fur- ther accuracy is required, additional points can be found with the aid of a calcu- lator and the use of symmetry properties. Given an equation of the form of equations (5) or (6), how can we find the coordinates of the foci without memorizing or looking up the relation b2 2 c2? There is a simple geometric relationship in an ellipse that enables us to get the same result using the Pythagorean theorem. To see this relationship, refer to Figure 4(a). Then, using the definition of an ellipse and 2a for the constant sum, as we did in deriving the standard equations, we see that — 2a a Thus, The length of the line segment from the end of a minor axis to a focus is the same as half the length of a major axis.
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-1 539 7-2 Ellipse This geometric relationship is illustrated in Figure 4(b). Using the Pythagorean theorem for the triangle in Figure 4(b), we have or Equations (4) and (7) or — b2 Useful for finding the foci, given a and b Thus, we can find the foci of an ellipse given the intercepts a and b simply by using the triangle in Figure 4(b) and the Pythagorean theorem. FIGURE q Geometric relationships. a>b>0 — 2a c (a) a (b) We summarize all of these results for convenient reference in Theorem 1. STANDARD EOURTIONS OF RN ELLIPSE WITH CENTER RT [O, O] THEOREM x intercepts: ±a (vertices) y intercepts: ±b Foci: F'(-c, 0), F(c, 0) a Major axis length Minor axis length c - 2b
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540 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY -1 Foci: F'(0, -c), F(0, c) —b —a - 10 -1 THEOREM continued Explore/ DI scuss E HRM PLE Solutions 2 b2 2 a x intercepts: y intercepts: a 2 2 ±a (vertices) — b2 a x b Major axis length = 2a Minor axis length 2b [Note: Both graphs are symmetric with respect to the x axis, y axis, and origin. Also, the major axis is always longer than the minor axis.] The line through a focus F of an ellipse that is perpendicular to the major axis intersects the ellipse in two points G and H. For each of the two standard equations of an ellipse with center (0, 0), find an expression in terms of a and b for the distance from G to H. Graphing Ellipses Sketch the graph of each equation, find the coordinates of the foci, and find the lengths of the major and minor axes. Check by graphing on a graphing utility. (A) 9x2 + 16y2 (B) 2x2 + (A) First, write the equation in standard form by dividing both sides by 144: 9x2 + 16y2 9X2 16y2 144 x2 Y2 16 9 144 144 2 — 16 and b2 Locate the intercepts: x intercepts: y intercepts: ± 3
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541 FIGURE S 9X2 + = 144. 7-2 Ellipse and sketch in the ellipse, as shown in Figure 5. — b2 _ 16 —4 —c —4.5 - 10 -5 5 — 10 2 Foci: c 2 3 -3 4 c x 4 FIGURE 6 (144 — 9x2)/16; (144 — 9x2)/16. 3 -3 FIGURE ? 4.5 c is positive. vfi, 0) and F(Vä, 0). Thus, the foci are F' (— Major axis length -8 Minor axis length - 2(3) = 6 To check the graph on a graphing utility, we solve the original equation for — 144 — (144 — 9x2)/16 This produces the two functions whose graphs are shown in Figure 6. 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 a common occurrence in graphs involving the square root function. (B) Write the equation in standard form by dividing both sides by 10: 2x2 + 10 10 10 5 10 10 2 — 10 and b2 2x2 + Y2 = 10. 10 x 5 Locate the intercepts: x intercepts: ± V/S ± 2.24 y intercepts: ±3.16 and sketch in the ellipse, as shown in Figure 7. 10 c 10 F 2 Foci: c 2 — b2 5
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542 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY — 52 — 16 FIGURE 8 10 - 2x2•, 10 - 2x2. 4 —4 MATCHED PROBLEM E HRM PLE 2 Solutions FIGURE S 36 100 6 VS) and F(0, VS). Thus, the foci are F' (0, Major axis length - 6.32 — 2VS 4.47 Minor axis length Figure 8 shows a check of the graph. Sketch the graph of each equation, find the coordinates of the foci, and find the lengths of the major and minor axes. Check by graphing on a graphing utility. (B) 3x2 + - 18 Finding the Equation of an Ellipse Find an equation of an ellipse in the form -1 if the center is at the origin, the major axis is along the y axis, and (A) Length of major axis = 20 Length of minor axis - 12 (B) Length of major axis - 10 Distance of foci from center = 4 Compute x and y intercepts and make a rough sketch of the ellipse, as shown -10 (B) in Figure 9. 10 -10 b2 a x 10 2 x -1 2 a 20 - 10 2 12 b -6 2 -1 36 100 —b FIGURE 10 2 x 9 25 5 4 5 x b —5 Make a rough sketch of the ellipse, as shown in Figure 10; locate the foci and y intercepts, then determine the x intercepts using the special triangle relationship discussed earlier. b2 a 9 -1 2 a 10 2 -1 25 b2 b — 42 — 25
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543 MATCHED PROBLEM 2 Explore/ DI scuss FIGURE Il Uses of elliptical forms. Planet Sun Planetary motion (a) 7-2 Ellipse Find an equation of an ellipse in the form x2 Y2 0 if the center is at the origin, the major axis is along the x axis, and (A) Length of major axis Length of minor axis - 50 - 30 (B) Length of minor axis - 16 Distance of foci from center -6 Consider the graph of an equation in the variables x and y. The equation of its magnification by a factor k > 0 is obtained by replacing x and y in the equation by x/k and y/k, respectively. (A) Find the equation of the magnification by a factor 3 of the ellipse with equation (x2/4) + Y2 — l. Graph both equations. (B) Give an example of an ellipse with center (0, 0) with a > b that is not a magnification of (x2/4) + y 2 (C) Find the equations of all ellipses that are magnifications of (x2/4) + Y2 Rpphcatlons You are no doubt aware of many occurrences and uses of elliptical forms: orbits of satellites, planets, and comets; shapes of galaxies; gears and cams; some air- plane wings, boat keels, and rudders; tabletops; public fountains; and domes in buildings are a few examples (see Fig. Il). Elliptical gears (b) Elliptical dome (c) Johannes Kepler (1571—1630), 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 [Fig. Il(a)]. Figure Il(b) shows a pair of elliptical gears with pivot points at foci. Such gears transfer constant rotational speed to variable rota- tional speed, and vice versa. Figure Il(c) shows an elliptical dome. An interest- ing property 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
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544 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY E HRM PLE 3 FIGURE Lithotripter. Solution MATCHED PROBLEM 3 Building in Washington, D.C., has such a dome, and is referred to as a whisper- ing room because a whispered sound at one focus can be easily heard at the other focus. A fairly recent application in medicine is the use of elliptical reflectors and ultrasound to break up kidney stones. A device called a lithotripter is used to gen- erate intense sound waves that break up the stone from outside the body, thus avoiding surgery. To be certain that the waves do not damage other parts of the body, the reflecting property of the ellipse is used to design and correctly posi- tion the lithotripter. Medicinal Lithotripsy A lithotripter is formed by rotating the portion of an ellipse below the minor axis around the major axis (see Fig. 12). The lithotripter is 20 centimeters wide and 16 centimeters deep. If the ultrasound source is positioned at one focus of the ellipse and the kidney stone at the other, then all the sound waves will pass through the kidney stone. How far from the kidney stone should the point V on the base of the lithotripter be positioned to focus the sound waves on the kidney stone? Round the answer to one decimal place. Ultrasound source Base Kidney stone 20 cm 16 cm From the figure, 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 kidney stone or the ultrasound source is given by 2 a 156 12.5 and the distance from the base of the lithotripter to the kidney stone is 16 + 12.5 — 28.5 centimeters. Since lithotripsy is an external procedure, the lithotripter described in Example 3 can be used only on stones within 12.5 centimeters of the surface of the body. Suppose a kidney stone is located 14 centimeters from the surface. If the diame- ter is kept fixed at 20 centimeters, how deep must a lithotripter be to focus on this kidney stone? Round answer to one decimal place.
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7-2 Ellipse Rnsuers -2 —5 —5 = 12 —5 to Matched Problems y 1 2 +21=1 x 625 225 EXERCISE 7-2 Foci: F'(-VS, 0), F(VS, 0) Major axis length = 4 Minor axis length = 2 (B) x 2 100 3. 18 F 6 18 17.2 centimeters Foci: F'(O, 12), (0, 12) Major axis length = 2 18 8.49 Minor axis length = 2 VC 4.90 x 6 64 In Problems 1—6, sketch a graph of each equation, find the coordinates of the foci, and find the lengths of the major and minor axes. Check by graphing on a graphing utility. —5 5 —5 (c) x 5 5 (d) 545 x 5 1. 25 4 2 4 25 9 4 4 9 In Problems 7—10, match each equation with one of graphs In Problems 11—16, sketch a graph of each equation, find the coordinates of the foci, and find the lengths of the major and minor axes. Check by graphing on a graphing utility. 7. 9x2 + 16y 2 = 16 5 —5 —5 (a) 8. 16x2 + 9y, 2=16 5 —5 (b) 11. 12. 13. 14. 15. 16. 25x2 + 9y2 = 225 16x2 + 25y2 = 400 2x2 -k y 2 4x2 + 3 Y2 4x2 + 7y2 3x2 -k 2)' 2 = 24 28 = 24 x 5 x 5 In Problems 17—28, find an equation of an ellipse in the form 2 x if the center is at the origin, and
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17. The graph is 10 -10 -10 18. The graph is 10 -10 -10 19. The graph is 10 -10 -10 20. The graph is 10 -10 -10 21. Major axis on x axis Major axis length = 10 Minor axis length = 6 546 1— 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY = 10 63 33 12y=O x 10 x 10 x 10 x 10 22. 23. 24. 25. 26. 27. 28. 29. 30. Major axis on x axis Major axis length = 14 Minor axis length = 10 Major axis on y axis Major axis length = 22 Minor axis length = 16 Major axis on y axis Major axis length = 24 Minor axis length = 18 Major axis on x axis Major axis length = 16 Distance of foci from center = 6 Major axis on y axis Major axis length = 24 Distance of foci from center Major axis on y axis Minor axis length 20 Distance of foci from center Major axis on x axis Minor axis length = 14 Distance of foci from center Explain why an equation whose graph is an ellipse does not define a function. Consider all ellipses having (0, ± 1) as the ends of the mi- nor axis. Describe the connection between the elongation of the ellipse and the distance from a focus to the origin. _LL In Problems 31 —38, find all points of intersection. Round any approximate values to three decimal places. 31. 35. 37. 39. 40. 16x2 + 25y2 = 400 2x- 5y = 10 33. 25x2 + 16y2 = 400 25x2 - 36y = O 32. 25x2 + 16y2 400 5x + 8y = 20 16x2 + 25y2 = 400 3x2 20y = O 5x2 -k 2)' 2 2x-y=O 2x2 -k 3)' 2 x2 8y 34. 36. 38. 3X2 + x — 2y 3x2 -k 2)' 2 2 x = 57 = 43 Find an equation of the set of points in a plane, each of whose distance from (2, 0) is one-half its distance from the line x = 8. Identify the geometric figure. Find 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 y = 16. Identify the geometric figure.
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547 In Problems 41 —44, find the coordinates of all points of intersection to two decimal places. 41. X2 + 3)' 2 = 20,4x+ 5y = 11 44. 2x2+ 7y2 = 95, 13x2 + 6y2 42. 8x2 + 35y2 = 3,600, 43. -25y 7-2 Ellipse Leading edge Trailing edge 2 = 1,025, 9x2 + 2y2 = 300 = 63 APPLICATIONS 45. Engineering. The semielliptical arch in the concrete bridge in the figure must have a clearance of 12 feet above the water and span a distance of 40 feet. Find the equation of the ellipse after inserting a coordinate system with the center of the ellipse at the origin and the major axis on the x axis. The y axis points up, and the x axis points to the right. How much clearance above the water is there 5 feet from the bank? Elliptical bridge 46. Design. A 4 >< 8 foot elliptical tabletop is to be cut out of a 4 >< 8 foot rectangular sheet of teak plywood (see the figure). To draw the ellipse on the plywood, 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 Fig. 1 in the text)? Compute the answer to two decimal places. String Elliptical table * 47. 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 may be a straight line, as shown in the figure. One of the most famous planes with this design was the World War Il British Spitfire. The plane in the figure has a wingspan of 48.0 feet. Fuselage Elliptical wings and tail * 48. (A) If the straight-line leading edge is parallel to the major axis 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 x axis lie along the major axis (positive right), and let the y axis lie along the minor axis (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. Naval Architecture. Currently, many high-performance racing sailboats use elliptical keels, rudders, and main sails for the same reasons stated in Problem 47—1ess drag along the trailing edge. In the accompanying figure, the el- lipse containing the keel has a 12.0-foot major axis. The straight-line leading edge is parallel to the major axis of the ellipse and 1.00 foot in front of it. The chord is 1.00 foot shorter than the major axis. IDA Rudder Keel (A) Find the equation of the ellipse. Let the y axis lie along the minor axis of the ellipse, and let the x axis lie along the major axis, both with positive direction upward. (B) What is the width of the keel, measured perpendicular to the major axis, 1 foot up the major axis from the bottom end of the keel? Compute quantities to three significant digits.

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