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Match Column-I with the correct numerical values: (A) Number of positive integral values of b for which a tangent parallel to y = x + 1 can be drawn to the hyperbola x²/5 - y²/b² = 1. (B) The hyperbola with vertices (3,0), (-3,0) and semi-latus rectum 4 is 4x² - 3y² = 4k; find k. (C) The product of the perpendicular distances from the two foci to any tangent of x²/25 - y²/3 = 1 is sqrt(k); find k. (D) A tangent to 16x² - 25y² - 96x + 100y - 356 = 0 making angle pi/4 with the transverse axis is y = x + lambda (lambda > 0); find 2*lambda. Which set of values matches (A),(B),(C),(D)?

1. (A) infinite, (B) 9, (C) 9, (D) 4
2. (A) 2, (B) 9, (C) 9, (D) 16
3. (A) 16, (B) 2, (C) 4, (D) 9
4. (A) 4, (B) 9, (C) 9, (D) 2

Correct answer: (A) infinite, (B) 9, (C) 9, (D) 4

Solution

(B) b² = 4a = 12, so 4x² - 3y² = 4k with the hyperbola x²/9 - y²/12 = 1 gives k = 9. (C) product of focal perpendiculars = b² = 3 = sqrt(9), k = 9. (A) infinitely many integer b allow a slope-1 tangent. (D) gives 2*lambda = 4.

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