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For the hyperbola H: x²/a² - y²/b² = 1 (a > b > 0), the conjugate axis LM subtends 60 deg at a vertex N, and the area of triangle LMN is 4*sqrt(3). Match each quantity to its value. P: length of conjugate axis; Q: eccentricity; R: distance between foci; S: length of latus rectum. Choose the correct matching.
- P -> 4; Q -> 2/sqrt(3); R -> 8; S -> 4/sqrt(3)
- P -> 8; Q -> 2; R -> 4; S -> 2/sqrt(3)
- P -> 4; Q -> sqrt(3); R -> 4; S -> 2
- P -> 2; Q -> 2/sqrt(3); R -> 4; S -> 8
Correct answer: P -> 4; Q -> 2/sqrt(3); R -> 8; S -> 4/sqrt(3)
Solution
tan30 = b/a gives a = sqrt(3) b; area ab = 4sqrt3 yields b = 2, a = 2sqrt3, from which all four quantities follow.
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