Consider the hyperbola with foci at and , where lies on the positive -axis. Let be a point on the hyperbola, in the first quadrant. Let with . The straight line passing through the point and having the same slope as that of the tangent at to the hy

Question

Consider the hyperbola $\frac{x^{2}}{100} - \frac{y^{2}}{64} = 1$ 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 $α < \frac{π}{2}$ . 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 $δ$ be the distance of $P$ from the straight line $S P_{1}$ and $β = S_{1} P .$ Then the greatest integer less than or equal to $\frac{β δ}{9} \sin \frac{α}{2}$ is ________. [2022]

Smart Achievers · Accepted Answer

(7)

In $∆ S_{1} Q P,$ $\sin \frac{α}{2} = \frac{S_{1} Q}{β}$

$\Rightarrow S_{1} Q = β \sin \frac{α}{2}$

Product of distances of any tangent from two foci $= b^{2}$

$δ \cdot S_{1} Q = δ \times β \sin \frac{α}{2} = b^{2}$

So, $\frac{β δ \sin \frac{α}{2}}{9} = \frac{b^{2}}{9} = \frac{64}{9}$

$∴$ $[\frac{64}{9}] = 7_{s s}$

Solution

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Solution

Ans. (7) In ∆S1QP, sinα2=S1Qβ ⇒S1Q=βsinα2 Product of distances of any tangent from two foci =b2 δ·S1Q=δ×βsinα2=b2 So, βδsinα29=b29=649 ∴ [649]=7ss

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