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 slope…

Question

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 _____ .

Aicharya · Accepted Answer

Plotting the diagram of the given value we have,
We know that product of distances of any tangent from two foci $= b^{2}$
So, $δ \times β \sin \frac{α}{2} = b^{2}$
$\Rightarrow \frac{β δ \sin \frac{α}{2}}{9} = \frac{b^{2}}{9} = \frac{64}{9}$
Now finding the greatest integer less than or equal to $\frac{β δ \sin \frac{α}{2}}{9}$ which will be $7$

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.…

Solution