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If two points P and Q lie on the hyperbola x 2 a 2 - y 2 b 2 = 1 ( a < b ) , whose centre C be such that C P is perpendicular to C Q , then the value of 1 C P 2 + 1 C Q 2 is

Options

  1. Ab 2 - a 2 2 a b
  2. B1 a 2 + 1 b 2
  3. C2 a b b 2 - a 2
  4. D1 a 2 - 1 b 2

Correct answer

D. 1 a 2 - 1 b 2

Step-by-step solution

Let C P = r 1 be inclined to transverse axis at an angle &#952; such that P r 1 cos &#8289; &#952; , &#160; r 1 sin &#8289; &#952; lies on the hyperbola. Therefore, r 1 2 &#160; cos 2 &#8289; &#952; a 2 - sin 2 &#8289; &#952; b 2 = 1 Replacing &#952; &#160; b y &#160; 90 o + &#952; , we get, r 2 2 &#160; sin 2 &#8289; &#952; a 2 - cos 2 &#8289; &#952; b 2 = 1 (where, C Q = r 2 ) &#8658; &#160; &#160; 1 r 1 2 + 1 r 2 2 = cos 2 &#8289; &#952; a 2 &#160; - sin 2 &#8289; &#952; b 2 + sin 2 &#8289; &#952; a 2 - cos 2 &#

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