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Consider the hyperbola 2x² - 2y² - 4x + 41 = 0. If there are exactly two points on this hyperbola from which perpendicular tangents can be drawn to the ellipse x²/a² + y²/b² = 1 (where a, b are natural numbers), find |a - b|.
- 2
- 4
- 6
- None of these
Correct answer: 2
Solution
The director circle x²+y²=a²+b² and the hyperbola (x-1)²-y² = -39/2 intersect at exactly 2 points when the discriminant of the resulting quadratic is zero, giving a²+b²=20. The only natural-number solutions are (2,4) and (4,2), both giving |a-b|=2.
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