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An ellipse, with axes along the coordinate axes, cuts the hyperbola 2x² - 2y² = 1 orthogonally, and its eccentricity is the reciprocal of the hyperbola's eccentricity. Which of the following is correct?
- Equation of the ellipse is x² + 2y² = 2
- The foci of the ellipse are (±1, 0)
- Equation of the ellipse is x² + 2y² = 4
- The foci of the ellipse are (±sqrt(2), 0)
Correct answer: Equation of the ellipse is x² + 2y² = 2
Solution
The rectangular hyperbola has e = sqrt(2), so the ellipse has e = 1/sqrt(2), giving B = A/2; the orthogonal-intersection condition yields A = 2, i.e. x² + 2y² = 2.
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