Section 7-2 Ellipse. Definition of an Ellipse The following is a coordinate-free definition of an ellipse: DEFINITION

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1 7- Ellipse 3. 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. (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. Signal light ocus Section 7- 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 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. V d 1 d Constant B d 1 P d B V

2 36 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Drawing an Ellipse An ellipse is eas to draw. All ou need is a piece of string, two thumbtacks, and a pencil or pen (see 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. 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. 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 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 coordinate sstem. We start b placing an ellipse in the coordinate sstem with the foci on the ais equidistant from the origin at (c, 0) and (c, 0), as in igure. IGURE Ellipse with foci on ais. P(, ) d 1 d (c, 0) 0 (c, 0) d 1 d Constant c 0 or reasons that will become clear soon, it is convenient to represent the constant sum d 1 d b a, a 0. Also, the geometric fact that the sum of the lengths of an two sides of a triangle must be greater than the third side can be applied to igure to derive the following useful result: d(, P) d(p, ) d(, ) d 1 d c a c a c (1)

3 7- Ellipse 37 We will use this result in the derivation of the equation of an ellipse, which we now begin. Referring to igure, the point P(, ) is on the ellipse if and onl if d 1 d a d(p, ) d(p, ) a ( c) ( 0) ( c) ( 0) a After eliminating radicals and simplifing, a good eercise for ou, we obtain (a c ) a a (a c ) a a c 1 () (3) Dividing both sides of equation () b a (a c ) is permitted, since neither a nor a c is 0. rom equation (1), a c; thus a c and a c 0. The constant a was chosen positive at the beginning. To simplif equation (3) further, we let b a c b 0 (4) to obtain a b 1 () rom equation () we see that the intercepts are a and the intercepts are b. The intercepts are also the vertices. Thus, Major ais length a Minor ais length b To see that the major ais is longer than the minor ais, we show that a b. Returning to equation (4), b a c b c a b a b a 0 (b a)(b a) 0 b a 0 b a b a a b Length of major ais Length of minor ais a, b, c 0 Definition of Since b a is positive, b a must be negative.

4 38 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY If we start with the foci on the ais at (0, c) and (0, c) as in igure 3, instead of on the ais as in igure, then, following arguments similar to those used for the first derivation, we obtain b a 1 a b (6) where the relationship among a, b, and c remains the same as before: b a c (7) The center is still at the origin, but the major ais is now along the ais and the minor ais is along the ais. IGURE 3 Ellipse with foci on ais. (0, c) d 1 P(, ) 0 d (0, c) d 1 d Constant c 0 To sketch graphs of equations of the form of equations () or (6) is an eas matter. We find the and intercepts and sketch in an appropriate ellipse. Since replacing with or with produces an equivalent equation, we conclude that the graphs are smmetric with respect to the ais, ais, and origin. If further accurac is required, additional points can be found with the aid of a calculator and the use of smmetr properties. Given an equation of the form of equations () or (6), how can we find the coordinates of the foci without memorizing or looking up the relation b a c? There is a simple geometric relationship in an ellipse that enables us to get the same result using the Pthagorean theorem. To see this relationship, refer to igure 4(a). Then, using the definition of an ellipse and a for the constant sum, as we did in deriving the standard equations, we see that Thus, d d a d a d a The length of the line segment from the end of a minor ais to a focus is the same as half the length of a major ais.

5 7- Ellipse 39 This geometric relationship is illustrated in igure 4(b). Using the Pthagorean theorem for the triangle in igure 4(b), we have or b c a b a c Equations (4) and (7) or c a b Useful for finding the foci, given a and b Thus, we can find the foci of an ellipse given the intercepts a and b simpl b using the triangle in igure 4(b) and the Pthagorean theorem. IGURE 4 Geometric relationships. b a b 1 a b 0 b a b c a d c 0 c a d a c 0 c a a b b (a) (b) We summarize all of these results for convenient reference in Theorem 1. THEOREM 1 STANDARD EQUATIONS O AN ELLIPSE WITH CENTER AT (0, 0) 1. a b 0 a b 1 intercepts: a (vertices) intercepts: b oci: (c, 0), (c, 0) c a b a b a c 0 c a Major ais length a Minor ais length b b

6 40 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY THEOREM 1 continued. a b 0 b a 1 intercepts: b intercepts: a (vertices) oci: (0, c), (0, c) a c a c a b b 0 b Major ais length a Minor ais length b [Note: Both graphs are smmetric with respect to the ais, ais, c a and origin. Also, the major ais is alwas longer than the minor ais.] Eplore/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 Ellipses Sketch the graph of each equation, find the coordinates of the foci, and find the lengths of the major and minor aes. Check b graphing on a graphing utilit. (A) (B) Solutions (A) irst, write the equation in standard form b dividing both sides b 144: a 16 and b 9 Locate the intercepts: intercepts: 4 intercepts: 3

7 7- Ellipse 41 4 c IGURE c 4 and sketch in the ellipse, as shown in igure. oci: c a b c 7 c is positive. 3 Thus, the foci are ( 7, 0) and ( 7, 0). Major ais length (4) 8 Minor ais length (3) 6 IGURE 6 1 (144 9 )/16; (144 9 )/16. 3 To check the graph on a graphing utilit, we solve the original equation for : (144 9 )/16 (144 9 )/ This produces the two functions whose graphs are shown in igure 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 b dividing both sides b : 1 a and b IGURE 7. c Locate the intercepts: intercepts:.4 intercepts: 3.16 and sketch in the ellipse, as shown in igure 7. c 0 oci: c a b c

8 4 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY IGURE 8 1 ;. 4 Thus, the foci are (0, ) and (0, ). Major ais length 6.3 Minor ais length igure 8 shows a check of the graph. 4 MATCHED PROBLEM 1 Sketch the graph of each equation, find the coordinates of the foci, and find the lengths of the major and minor aes. Check b graphing on a graphing utilit. (A) 4 4 (B) 3 18 EXAMPLE inding the Equation of an Ellipse ind an equation of an ellipse in the form M N 1 M, N 0 if the center is at the origin, the major ais is along the ais, and (A) Length of major ais 0 (B) Length of major ais Length of minor ais 1 Distance of foci from center 4 Solutions (A) Compute and intercepts and make a rough sketch of the ellipse, as shown in igure 9. IGURE b a a b IGURE 1. 9 (B) Make a rough sketch of the ellipse, as shown in igure ; locate the foci and intercepts, then determine the intercepts using the special triangle relationship discussed earlier. b 4 0 b b a 1 a b b 3 9 1

9 7- Ellipse 43 MATCHED PROBLEM ind an equation of an ellipse in the form M N 1 M, N 0 if the center is at the origin, the major ais is along the ais, and (A) Length of major ais 0 (B) Length of minor ais 16 Length of minor ais 30 Distance of foci from center 6 Eplore/Discuss Consider the graph of an equation in the variables and. The equation of its magnification b a factor k 0 is obtained b replacing and in the equation b /k and /k, respectivel. (A) ind the equation of the magnification b a factor 3 of the ellipse with equation ( /4) 1. Graph both equations. (B) Give an eample of an ellipse with center (0, 0) with a b that is not a magnification of ( /4) 1. (C) ind the equations of all ellipses that are magnifications of ( /4) 1. Applications You are no doubt aware of man occurrences and uses of elliptical forms: 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 (see ig. 11). IGURE 11 Uses of elliptical forms. Planet Sun Planetar motion Elliptical gears Elliptical dome (a) (b) (c) 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. 11(a)]. igure 11(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 11(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

10 44 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY 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 3 Medicinal Lithotrips A lithotripter is formed b rotating the portion of an ellipse below the minor ais around the major ais (see ig. 1). 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. IGURE 1 Lithotripter. Ultrasound source Kidne stone Base V 0 cm 16 cm Solution rom the figure, we see that a 16 and b 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 3 Since lithotrips is an eternal procedure, the lithotripter described in Eample 3 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.

11 7- Ellipse 4 Answers to Matched Problems 1. (A) oci: (3, 0), (3, 0) (B) Major ais length 4 1 Minor ais length oci: (0, 1), (0, 1) Major ais length Minor ais length (A) (B) centimeters EXERCISE 7- A 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 aes. Check b graphing on a graphing utilit (c) (d) In Problems 7, match each equation with one of graphs (a) (d) (a) (b) B 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 aes. Check b graphing on a graphing utilit In Problems 17 8, find an equation of an ellipse in the form M, N 0 M N 1 if the center is at the origin, and

12 46 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY 17. The graph is. Major ais on ais Major ais length 14 Minor ais length 3. Major ais on ais Major ais length Minor ais length Major ais on ais Major ais length 4 Minor ais length The graph is. Major ais on ais Major ais length 16 Distance of foci from center 6 6. Major ais on ais Major ais length 4 Distance of foci from center 7. 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 The graph is 9. Eplain wh an equation whose graph is an ellipse does not define a function. 30. 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. 0. The graph is In Problems 31 38, find all points of intersection. Round an approimate values to three decimal places C 39. 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. 1. Major ais on ais Major ais length Minor ais length 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.

13 7- Ellipse 47 In Problems 41 44, find the coordinates of all points of intersection to two decimal places. Leading edge , ,600, ,0, , APPLICATIONS Elliptical wings and tail uselage Trailing edge 4. 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. 48. Naval Architecture. Currentl, man high-performance racing sailboats use elliptical keels, rudders, and main sails for the same reasons stated in Problem 47 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. 46. Design. A4 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 Rudder Keel 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 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. (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.