Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. Let P $\left(3\, sec \,\theta, \,2 \,tan \,\theta\right)$ and Q $\left(3\, sec\, \phi, \,2 \,tan \,\phi\right)$ where $\theta + \phi = \frac{\pi}{2},$ be two distinct points on the hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1.$ Then the ordinate of the point of intersection of the normals at P and Q is :

JEE MainJEE Main 2014Conic Sections

Solution:

$p \left(3 \,sec\, \theta,\,2 \,tan \,\theta\right) \quad Q = \left(3 \,sec\,\phi\, ,\, 2 \,tan\phi\right)$
$\theta +\phi = \frac{\pi}{2}\quad Q = \left(3 \,cosec \, \theta , \,2 \, cot\theta\right)$
Equation of normal at $p =$
$= 3x \, cos \,\theta + 2y \, cot \,\theta=13$
$= 3x \, sin \,\theta \, cos \,\theta + 2y \,cos \,\theta = 13 \,sin\theta\quad ...\left(1\right)$
equation of normal at Q $\Rightarrow $
$= 3x\, sin \, \theta + 2y \, tan \, \theta = 13$
$= 3x\, sin\, \theta \, cos \, \theta + 2y \, sin \, \theta = 13\, cos\theta \quad ...\left(2\right)$
$\left(1\right)-\left(2\right)\, \Rightarrow $
$2y \, \left(cos\, \theta - sin \, \theta \right) = 13 \left(sin\, \theta - cos \,\theta \right)$
$2y = -13 \Rightarrow \frac{-13}{2}$