Consider the hyperbola x 2 100 - y 2 64 = 1 with foci at S and S 1 , where S lies on the positive x -axis. Let P be a point on the hyperbola, in the first quadrant. Let ∠ S P S 1 = α , with α < π 2 . The straight line passing through the point S and having the same slo

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Consider the hyperbola x 2 100 - y 2 64 = 1  with foci at S and S 1 , where S lies on the positive x -axis. Let P be a point on the hyperbola, in the first quadrant. Let &#8736; S P S 1 = &#945; , with &#945; &#60; &#960; 2 . The straight line passing through the point S and having the same slope as that of the tangent at P to the hyperbola, intersects the straight line S 1 P at P 1 . Let &#948; be the distance of P  from the straight line S P 1 , and &#946; = S 1 P . Then the greatest integer less than or equal to &#946; &#948; 9 sin &#945; 2 is _____ .

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Correct answer: A. A

Plotting the diagram of the given value we have, We know that product of distances of any tangent from two foci = b 2 So,  &#948; &#215; &#946; sin &#945; 2 = b 2 &#8658; &#946; &#948; sin &#945; 2 9 = b 2 9 = 64 9 Now finding the greatest integer less than or equal to &#946; &#948; sin &#945; 2 9  which will be  7

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