1007 10.2 Ellipses NOTE As suggested by the graphs in this section, an ellipse is symmetric with respect to its major axis, its minor axis, and its center. If a =b in the equation of an ellipse, then its graph is a circle. Eccentricity All conics can be characterized by one general definition. Conic A conic is the set of all points P1x, y2 in a plane such that the ratio of the distance from P to a fixed point and the distance from P to a fixed line is constant. Think: c 1 2 1-82d 2 = 16 and c 1 2 1102d 2 = 25 Prepare to complete the square for both x and y. 491x2 - 8x + 16 - 162 + 91y2 + 10y + 25 - 252 = -568 Complete the square. 491x2 - 8x + 162 - 784 + 91y2 + 10y + 252 - 225 = -568 Distributive property 491x - 422 + 91y + 522 = 441 Factor. Add 784 and 225 on each side. 1x - 4 2 9 + 1y + 522 49 = 1 Divide by 441. Multiply 491-162 = -784 and 91-252 = -225 EXAMPLE 5 Writing the Equation of an Ellipse in Standard Form Write the equation of the ellipse 49x2 - 392x + 9y2 + 90y = -568 in standard form. Give the center, vertices, and endpoints of the minor axis. SOLUTION 49x2 - 392x + 9y2 + 90y = -568 Given equation 491x2 - 8x 2 + 91y2 + 10y 2 = -568 Factor out 49, and factor out 9. This equation represents an ellipse with center 14, -52. The major axis is vertical because a2 = 49 is associated with y2. The vertices are located a distance of a = 7 units directly above and below the center 14, -52, at 14, 22 and 14, -122. See Figure 22. Because b2 = 9 is associated with x2, the endpoints of the minor axis are located a distance of b = 3 units left and right of the center 14, -52, at 11, -52 and 17, -52, as shown in Figure 22. S Now Try Exercise 41. 49x2 – 392x + 9y2 + 90y = –568 + = 1 (y + 5)2 49 (x – 4)2 9 (1, –5) (7, –5) V(4, 2) V9(4, –12) y –5 –2 –12 2 1 4 7 x (4, –5) Figure 22 For a parabola, the fixed line is the directrix, and the fixed point is the focus. In Figure 23, the focus is F1c, 02, and the directrix is the line x = -c. The constant ratio is the eccentricity of the conic, written e. (This is not the same e as the base of natural logarithms.) If the conic is a parabola, then by definition, the distances d1P, F2 and d1P, D2 in Figure 23 are equal. Thus, every parabola has eccentricity 1. P(x, y) F(c, 0) D(–c, y) x = –c y 0 x e = d(P, F) d(P, D) Figure 23
RkJQdWJsaXNoZXIy NjM5ODQ=