1002 CHAPTER 10 Analytic Geometry As shown in Figure 14, an ellipse has two axes of symmetry, the major axis (the longer one) and the minor axis (the shorter one). The foci are always located on the major axis. The midpoint of the major axis is the center of the ellipse, and the endpoints of the major axis are the vertices of the ellipse. The graph of an ellipse is not the graph of a function. It fails the vertical line test. The ellipse in Figure 15 has its center at the origin, foci F1c, 02 and F′1-c, 02, and vertices V1a, 02 and V′1-a, 02. From Figure 15, the distance from V to F is a - c and the distance from V to F′ is a + c. The sum of these distances is 2a. Because V is on the ellipse, this sum is the constant referred to in the definition of an ellipse. Thus, for any point P1x, y2 on the ellipse, d1P, F2 + d1P, F′2 = 2a. By the distance formula, d1P, F2 = 21x - c22 + y2 and d1P, F′2 = 23x - 1-c242 + y2 = 21x + c22 + y2. Thus, we have the following. 21x - c22 + y2 + 21x + c22 + y2 = 2a d1P, F2 + d1P, F′2 = 2a 21x - c22 + y2 = 2a - 21x + c22 + y2 Isolate 21x - c22 + y2. 1x - c22 + y2 = 4a2 - 4a21x + c22 + y2 + 1x + c22 + y2 Square each side. x2 - 2cx + c2 + y2 = 4a2 - 4a21x + c22 + y2 + x2 + 2cx + c2 + y2 Square x - c. Square x + c. 4a21x + c22 + y2 = 4a2 + 4cx Isolate 4a21x + c22 + y2. a21x + c22 + y2 = a2 + cx Divide by 4. a21x2 + 2cx + c2 + y2 = a4 + 2ca2x + c2x2 Square each side. Square x + c. a2x2 + 2ca2x + a2c2 + a2y2 = a4 + 2ca2x + c2x2 Distributive property a2x2 + a2c2 + a2y2 = a4 + c2x2 Subtract 2ca2x. a2x2 - c2x2 + a2y2 = a4 - a2c2 Rearrange terms. 1a2 - c2 x2 + a2y2 = a21a2 - c22 Factor. x2 a2 + y2 a2 - c2 = 1 112 Divide by a21a2 - c22. x y Vertex Vertex Focus Focus Center Minor axis Major axis Figure 14 F'(–c, 0) F(c, 0) P(x, y) (0, b) (–a, 0) (a, 0) (0, –b) x y V' V B B' Figure 15 Ellipse centered at the origin Divide each term by 4. Be careful when squaring.
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