694 CHAPTER 10 Analytic Geometry Figure 21 x a y b a b 1, 0 2 2 2 2 + = > > x y (0, b) (0, 2b) b a c V1 5 (2a, 0) V2 5 (a, 0) F1 5 (2c, 0) F2 5 (c, 0) Finding an Equation of an Ellipse Find an equation of the ellipse with center at the origin, one focus at 3, 0 , ( ) and a vertex at 4, 0 . ( ) − Graph the equation. EXAMPLE 1 Figure 22 x y 16 7 1 2 2 + = V1 5 (24, 0) V2 5 (4, 0) x y 25 5 5 25 (0, 7 ) (0, 2 7 ) F2 5 (3, 0) F1 5 (23, 0) Because a c 0, > > this means a c , 2 2 > so a c 0. 2 2 − > Let b a c b , 0. 2 2 2 = − > Then a b > and equation (1) can be written as b x a y a b x a y b 1 2 2 2 2 2 2 2 2 2 2 + = + = Divide both sides by a b . 2 2 The graph of the equation has symmetry with respect to the x -axis, the y -axis, and the origin. Because the major axis is the x -axis, the vertices lie on the x -axis. So the vertices satisfy the equation x a 1, 2 2 = the solutions of which are x a. = ± Consequently, the vertices of the ellipse are V a, 0 1 ( ) = − and V a, 0 . 2 ( ) = The y -intercepts of the ellipse, found by substituting x 0 = in the equation, have coordinates b 0, ( ) − and b 0, . ( ) The four intercepts, a a b , 0 , , 0 , 0, , ( ) ( ) ( ) − and b 0, , ( ) − are used to graph the ellipse. THEOREM Equation of an Ellipse: Center at ( ) 0, 0 ; Major Axis along the x -Axis An equation of the ellipse with center at 0, 0 , ( ) foci at c, 0 ( ) − and c, 0 , ( ) and vertices at a, 0 ( ) − and a, 0 ( ) is x a y b a b b a c 1 where 0 and 2 2 2 2 2 2 2 + = > > = − (2) The major axis is the x -axis. See Figure 21. Notice in Figure 21 the points c 0,0, ,0, ( ) ( ) and b 0, ( ) form a right triangle. Because b a c b c a or , 2 2 2 2 2 2 ( ) = − + = the distance from the focus at c, 0 ( ) to the point b 0, ( ) is a. This can be seen another way. Look at the two right triangles in Figure 21. They are congruent. Do you see why? Because the sum of the distances from the foci to a point on the ellipse is a2 , it follows that the distance from c, 0 ( ) to b 0, ( ) is a. Solution The ellipse has its center at the origin, and since the given focus and vertex lie on the x -axis, the major axis is the x -axis. The distance from the center, 0, 0 , ( ) to one of the foci, 3, 0 , ( ) is c 3. = The distance from the center, 0, 0 , ( ) to one of the vertices, 4, 0 , ( ) − is a 4. = From equation (2), it follows that b a c 16 9 7 2 2 2 = − = − = so an equation of the ellipse is x y 16 7 1 2 2 + = Figure 22 shows the graph. In Figure 22, the intercepts of the equation are used to graph the ellipse. Following this practice makes it easier to obtain an accurate graph of an ellipse. Now Work PROBLEM 29
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