The second type of conic is called an ellipse, and is defined as follows. Definition of Ellipse

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1 72 Chapter 10 Topics in Analtic Geometr 10.3 ELLIPSES What ou should learn Write equations of ellipses in standard form and graph ellipses. Use properties of ellipses to model and solve real-life problems. Find eccentricities of ellipses. Wh ou should learn it Ellipses can be used to model and solve man tpes of real-life problems. For instance, in Eercise 65 on page 79, an ellipse is used to model the orbit of Halle s comet. Introduction The second tpe of conic is called an ellipse, and is defined as follows. Definition of Ellipse An ellipse is the set of all points, in a plane, the sum of whose distances from two distinct fied points foci) is constant. See Figure Focus d1, ) d2 Focus Major ais Center Minor ais Harvard College Observator/Photo Researchers, Inc. d 1 d 2 is constant. FIGURE FIGURE The line through the foci intersects the ellipse at two points called vertices. The chord joining the vertices is the major ais, and its midpoint is the center of the ellipse. The chord perpendicular to the major ais at the center is the minor ais of the ellipse. See Figure You can visualize the definition of an ellipse b imagining two thumbtacks placed at the foci, as shown in Figure If the ends of a fied length of string are fastened to the thumbtacks and the string is drawn taut with a pencil, the path traced b the pencil will be an ellipse., ) FIGURE b + c b + c h, k) 2 2 b To derive the standard form of the equation of an ellipse, consider the ellipse in Figure with the following points: center, h, k; vertices, h ± a, k; foci, h ± c, k. Note that the center is the midpoint of the segment joining the foci. The sum of the distances from an point on the ellipse to the two foci is constant. Using a verte point, this constant sum is 2 b 2 + c 2 = 2a b 2 + c 2 = FIGURE c a a c a c 2a Length of major ais or simpl the length of the major ais. Now, if ou let, be an point on the ellipse, the sum of the distances between, and the two foci must also be 2a.

2 Section 10.3 Ellipses 73 That is, h c 2 k 2 h c 2 k 2 2a which, after epanding and regrouping, reduces to c 2 k 2 c 2. Finall, in Figure 10.22, ou can see that b 2 c 2 which implies that the equation of the ellipse is b 2 k 2 b 2 k2 b 2 1. You would obtain a similar equation in the derivation b starting with a vertical major ais. Both results are summarized as follows. Consider the equation of the ellipse k2 b 2 1. If ou let a b, then the equation can be rewritten as k 2 which is the standard form of the equation of a circle with radius r a see Section 1.2). Geometricall, when a b for an ellipse, the major and minor aes are of equal length, and so the graph is a circle. Standard Equation of an Ellipse The standard form of the equation of an ellipse, with center h, k and major and minor aes of lengths 2 a and 2b, respectivel, where 0 < b < a, is Figure shows both the horizontal and vertical orientations for an ellipse. b 2 Major ais is horizontal. Major ais is vertical. The foci lie on the major ais, c units from the center, with c 2 b 2. If the center is at the origin 0, 0, the equation takes one of the following forms. 2 2 b 2 1 k2 b 2 1 k2 1. Major ais is horizontal. 2 b Major ais is vertical. h) 2 k) 2 + = 1 a2 b 2 h) 2 k) 2 + = 1 b2 a2 h, k) 2b h, k) 2a 2a 2b Major ais is horizontal. FIGURE Major ais is vertical.

3 7 Chapter 10 Topics in Analtic Geometr Eample 1 Finding the Standard Equation of an Ellipse FIGURE , 1) 2, 1), 1) 1 3 a = 3 b = 5 Find the standard form of the equation of the ellipse having foci at 0, 1 and, 1 and a major ais of length 6, as shown in Figure Because the foci occur at 0, 1 and, 1, the center of the ellipse is 2, 1) and the distance from the center to one of the foci is c 2. Because 2a 6, ou know that a 3. Now, from c 2 b 2, ou have b c Because the major ais is horizontal, the standard equation is This equation simplifies to Now tr Eercise 23. Eample 2 Sketching an Ellipse + 3) 2 1) 2 + = , 1) 3, 2) 1, 1) 2 3 3, 1) 3, 1) , 1) 5 3 3, 0) 1 1 FIGURE Sketch the ellipse given b Begin b writing the original equation in standard form. In the fourth step, note that 9 and are added to both sides of the equation when completing the squares Write original equation. Group terms Factor out of -terms. Write in completed square form. Divide each side b. Write in standard form. From this standard form, it follows that the center is h, k 3, 1. Because the denominator of the -term is 2 2, the endpoints of the major ais lie two units to the right and left of the center. Similarl, because the denominator of the -term is b 2 1 2, the endpoints of the minor ais lie one unit up and down from the center. Now, from c 2 b 2, ou have c So, the foci of the ellipse are 3 3, 1 and 3 3, 1. The ellipse is shown in Figure Now tr Eercise

4 Section 10.3 Ellipses 75 Eample 3 Analzing an Ellipse 1) 2 + 2) + 2 = , Focus 2 2 1, 2) 1, FIGURE Center Focus 1, 2) 1, 6) Find the center, vertices, and foci of the ellipse B completing the square, ou can write the original equation in standard form Write original equation. Group terms. The major ais is vertical, where h 1, k 2, a, b 2, and c b So, ou have the following Factor out of -terms. Write in completed square form. Divide each side b 16. Write in standard form. Center: 1, 2 Vertices: 1, 6 Foci: 1, , 2 1, 2 23 The graph of the ellipse is shown in Figure Now tr Eercise TECHNOLOGY You can use a graphing utilit to graph an ellipse b graphing the upper and lower portions in the same viewing window. For instance, to graph the ellipse in Eample 3, first solve for to get and Use a viewing window in which 6 9 and 7 3. You should obtain the graph shown below

5 76 Chapter 10 Topics in Analtic Geometr Application Ellipses have man practical and aesthetic uses. For instance, machine gears, supporting arches, and acoustic designs often involve elliptical shapes. The orbits of satellites and planets are also ellipses. Eample investigates the elliptical orbit of the moon about Earth. Eample An Application Involving an Elliptical Orbit 767,60 km Moon The moon travels about Earth in an elliptical orbit with Earth at one focus, as shown in Figure The major and minor aes of the orbit have lengths of 768,800 kilometers and 767,60 kilometers, respectivel. Find the greatest and smallest distances the apogee and perigee, respectivel) from Earth s center to the moon s center. Earth Perigee FIGURE ,800 km Apogee Because 2a 768,800 and 2b 767,60, ou have a 38,00 and b 383,820 which implies that c b 2 38, , ,108. So, the greatest distance between the center of Earth and the center of the moon is WARNING / CAUTION Note in Eample and Figure that Earth is not the center of the moon s orbit. a c 38,00 21,108 05,508 kilometers and the smallest distance is a c 38,00 21, ,292 kilometers. Now tr Eercise 65. Eccentricit One of the reasons it was difficult for earl astronomers to detect that the orbits of the planets are ellipses is that the foci of the planetar orbits are relativel close to their centers, and so the orbits are nearl circular. To measure the ovalness of an ellipse, ou can use the concept of eccentricit. Definition of Eccentricit The eccentricit e of an ellipse is given b the ratio e c a. Note that 0 < e < 1 for ever ellipse.

6 Section 10.3 Ellipses 77 To see how this ratio is used to describe the shape of an ellipse, note that because the foci of an ellipse are located along the major ais between the vertices and the center, it follows that 0 < c < a. For an ellipse that is nearl circular, the foci are close to the center and the ratio ca is small, as shown in Figure On the other hand, for an elongated ellipse, the foci are close to the vertices and the ratio ca is close to 1, as shown in Figure Foci Foci e = c a c e is small. e = c a c e is close to 1. a FIGURE FIGURE a NASA The time it takes Saturn to orbit the sun is about 29. Earth ears. The orbit of the moon has an eccentricit of e 0.059, and the eccentricities of the eight planetar orbits are as follows. Mercur: e Venus: Earth: e e Jupiter: Saturn: Uranus: e 0.08 e e Mars: e Neptune: e CLASSROOM DISCUSSION Ellipses and Circles a. Show that the equation of an ellipse can be written as k e 2 b. For the equation in part a), let a, h 1, and k 2, and use a graphing utilit to graph the ellipse for e 0.95, e 0.75, e 0.5, e 0.25, and e 0.1. Discuss the changes in the shape of the ellipse as e approaches 0. c. Make a conjecture about the shape of the graph in part b) when e 0. What is the equation of this ellipse? What is another name for an ellipse with an eccentricit of 0?

Inclination of a Line. Definition of Inclination

Inclination of a Line. Definition of Inclination 76 Chapter 0 Topics in Analtic Geometr 0. LINES What ou should learn Find the inclination of a line. Find the angle between two lines. Find the distance between a point and a line. Wh ou should learn it