10-2. Parabola, Ellipse, Hyperbola
hard

The vertices of a hyperbola $H$ are $(\pm 6,0)$ and its eccentricity is $\frac{\sqrt{5}}{2}$. Let $N$ be the normal to $H$ at a point in the first quadrant and parallel to the line $\sqrt{2} x + y =2 \sqrt{2}$. If $d$ is the length of the line segment of $N$ between $H$ and the $y$-axis then $d ^2$ is equal to $............$.

A

$215$

B

$216$

C

$217$

D

$218$

(JEE MAIN-2023)

Solution

$H : \frac{ x ^2}{36}-\frac{y^2}{9}=1$

equation of normal is $6 x \cos \theta+3 y \cot \theta=45$

$\text { slope }=-2 \sin \theta=-\sqrt{2}$

$\Rightarrow \theta=\frac{\pi}{4}$

Equation of normal is $\sqrt{2} x+y=15$

$P:(a \sec \theta, b \tan \theta)$

$\Rightarrow P (6 \sqrt{2}, 3)$ and $K (0,15)$

$d^2=216$

Standard 11
Mathematics

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