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The eccentricity of the hyperbola x²/a² - y²/b² = 1 is the reciprocal of the eccentricity of the ellipse x² + 4y² = 4. If the hyperbola passes through a focus of the ellipse, which statement is correct?
- the equation of the hyperbola is x²/3 - y²/2 = 1
- a focus of the hyperbola is (2, 0)
- the eccentricity of the hyperbola is sqrt(5/3)
- the equation of the hyperbola is x² - 3y² = 3
Correct answer: the equation of the hyperbola is x² - 3y² = 3
Solution
The ellipse has e = sqrt3/2 so the hyperbola has e = 2/sqrt3; it passes through (sqrt3, 0), giving a² = 3, b² = 1, i.e. x²/3 - y²/1 = 1, which is x² - 3y² = 3.
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