JEE Advanced 2022 — Paper 2 — MATH-I solution

JEE Advanced2022Paper 2MATH-I

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 ______.

Your answer

Solution

The line through $S$ parallel to the tangent at $P$ meets $S_{1} P$ at $P_{1}$ , and the configuration yields a right angle at $P_{1}$ between the focal chord $S_{1} P$ and the perpendicular distance from $P$ along $S P_{1}$ .

The tangent at $P$ bisects the external angle between the focal radii, so the angle between $S P_{1}$ and $S P$ equals $\frac{α}{2}$ . In the right triangle with hypotenuse $β = S_{1} P$ and the perpendicular from $S$ onto the tangent line, the geometric identity becomes $δ \cdot β sin \frac{α}{2} = b^{2}$ , where $b^{2} = 64$ is the product of the perpendicular distances from the two foci to any tangent of the hyperbola.

Hence $\frac{β δ sin ( α /2 )}{9} = \frac{64}{9} \approx 7.11$ , and the floor is $7$ .

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