Question

The equation to the straight line passing through the point $(a{\cos }^3\theta ,a{\sin }^3\theta )$ and perpendicular to the line $x\sec \theta +y\text{cosec}\theta =a$ is

• A. $x\cos \theta -y\sin \theta =a\cos 2\theta$
• B. $x\cos \theta +y\sin \theta =a\cos 2\theta$
• C. $x\sin \theta +y\cos \theta =a\cos 2\theta$
• D. None of these

Answer 1

Answer: (A)

$x\cos \theta -y\sin \theta =a\cos 2\theta$

Line perpendicular to $x\sec \theta +y\text{cosec}\theta =a$ is $x\text{cosec}\theta -y\sec \theta =\lambda$.
This line passes through the point $(a{\cos }^3\theta ,a{\sin }^3\theta ).$
Then,
$\left(a{\cos }^3\theta \right)cosec\theta -\left(a{\sin }^3\theta \right)sec\theta =\lambda$
$\Rightarrow \lambda =a(\frac{{\cos }^3\theta }{\sin \theta }-\frac{{\sin }^3\theta }{\cos \theta })$
$=a\frac{\cos 2\theta }{\cos \theta \sin \theta }$
Hence, the equation of line is $x\cos \theta -y\sin \theta =a\cos 2\theta$.