1005 10.2 Ellipses 0 x y + = 1 7 x2 16 y2 4 –4 Ë7 –Ë7 F(0, 3) F9(0, –3) Figure 19 EXAMPLE 2 Writing an Equation of an Ellipse Write an equation of the ellipse having center at the origin, foci at 10, 32 and 10, -32, and major axis of length 8 units. SOLUTION Because the major axis is 8 units long, 2a = 8 and thus a = 4. To find b2, use the relationship c2 = a2 - b2, with a = 4 and c = 3. c2 = a2 - b2 Relationship for ellipses 32 = 42 - b2 Let c = 3 and a = 4. 9 = 16 - b2 Apply the exponents. b2 = 7 Solve for b2. The foci are on the y-axis, so we use the larger value, a = 4, to find the denominator for y2, giving the equation in standard form. x2 7 + y2 16 = 1 Use x 2 b2 + y2 a2 = 1. A graph of this ellipse is shown in Figure 19. This relation has domain C -27, 27 D and range 3-4, 44. S Now Try Exercise 21. x y 0 5 4 –5 = 1 – x2 25 y 4 Figure 20 EXAMPLE 3 Graphing a Half-Ellipse Graph y 4 = B1 - x2 25 . Give the domain and range. SOLUTION We transform this equation to see that its graph is part of an ellipse. y 4 = B1 - x2 25 Given equation y2 16 = 1 - x2 25 Square each side. x2 25 + y2 16 = 1 Write in standard form. This is the equation of an ellipse with x-intercepts 1{5, 02 and y-intercepts 10, {42. In the original equation, the radical expression B1 - x2 25 represents a nonnegative number, so the only possible values of y are those that give the half-ellipse shown in Figure 20. This is the graph of a function with domain 3-5, 54 and range 30, 44. S Now Try Exercise 33. Translated Ellipses An ellipse may have its center translated away from the origin by replacing x and y with x - h and y - k, respectively.
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