Q. The equation of a straight line which passes through the point $(a\ \cos^{3}\theta,\ {a\ \sin}^{3}\theta)$ and perpendicular to $x\sec\theta + y\ cosec\ \theta = a$ is
A) $\frac{x}{a} + \frac{y}{a} = a\cos\theta$
B) $x\cos\theta - \ y\sin\theta = - a\cos{2\theta}$
C) $x\cos\theta + y\sin\theta = a\cos{2\theta}$
D) $x\cos\theta + y\sin\theta - a\cos{2\theta}$
Explanation
(b)
Required equation of line is
$y - a\ \sin^{3}\theta = \frac{cosec\theta}{\sec\theta}\left( x - a\cos^{3}\theta \right)$
$\Rightarrow y - a\ \sin^{3}\theta = \frac{cos\theta}{\sin\theta}\left( x - a\ \cos^{3}\theta \right)$
$\Rightarrow y\sin\theta - a\sin^{4}\theta = x\cos\theta - a\ \cos^{4}\theta$
$\Rightarrow x\cos\theta - y\sin\theta + a\sin^{4}\theta - a\ \cos^{4}\ \theta = 0$
$\Rightarrow x\cos\theta - \ y\sin\theta + a\cos{2\ \theta} = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.