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Let e1 be the eccentricity of the hyperbola x²/16 - y²/9 = 1 and e2 be the eccentricity of an ellipse x²/a² + y²/b² = 1 (a > b > 0) that passes through both foci of the hyperbola. Given that e1 * e2 = 1, find the length of the chord of this ellipse that is parallel to the x-axis and passes through the point (0, 2).
- 4*sqrt(5)
- 8*sqrt(5)/3
- 10*sqrt(5)/3
- 3*sqrt(5)
Correct answer: 10*sqrt(5)/3
Solution
The hyperbola has e1 = 5/4, giving e2 = 4/5. The ellipse passes through (+-5, 0) so a = 5. Then b² = 25*(1 - 16/25) = 9. Substituting y=2 into the ellipse equation x²/25 + y²/9 = 1 gives x = +-5*sqrt(5)/3, so the chord length is 10*sqrt(5)/3.
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