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Stitz-Zeager_College_Algebra_e-book

Graphing the lines x y 0 x y 2 and 2x y 2 we see

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Unformatted text preview: the center to a vertex In an ellipse, the foci are closer to the center than the vertices, so 0 < e < 1. For the ellipse above on the left, e ≈ 0.98; for the ellipse on the right, e ≈ 0.66. In general, the closer the eccentricity is to 0, the more ‘circular’ the ellipse; the closer the eccentricity is to 1, the more ‘eccentric’ the ellipse. 426 Hooked on Conics 1 Example 7.4.4. Find the equation of the ellipse whose vertices are (0, ±5) with eccentricity e = 4 . Solution. As before, we plot the data given to us y x From this sketch, we know that the major axis is vertical, meaning b > a. With the vertices located at (0, ±5), we get b = 5 so b2 = 25. We also know that the center is (0, 0) because the center is the midpoint of the vertices. All that remains is to find a2 . To that end, we use the fact that the 1 eccentricity e = 4 which means distance from the center to a focus c c 1 === distance from the center to a vertex b 5 4 √ 5 from which we get c = 4 . To get a2 , we use the fact that c = b2 − a2 e= 5 4 √ c= b2 − a2 √ 5 25 −...
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