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Consider the conic C: x²/16 + y²/12 = 1. A hyperbola passes through the endpoints of the minor axis of C, with its transverse and conjugate axes lying along the minor and major axes of C respectively, and the product of the eccentricities of the hyperbola and C equals 1. Which of the following is correct?

1. equation of hyperbola is x²/36 - y²/12 = -1
2. equation of hyperbola is x²/9 - y²/3 = -1
3. focus of hyperbola is (0, 2*sqrt(3))
4. focus of hyperbola is (0, -4*sqrt(3))

Correct answer: equation of hyperbola is x²/36 - y²/12 = -1

Solution

C has e = 1/2, so the hyperbola has e = 2; with conjugate semi-axis 2*sqrt(3) the hyperbola is x²/36 - y²/12 = -1 (foci at (0, +/-4*sqrt(3))).

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