Question

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

Answer 1

Given: Passing throgh the point
$(a{\cos }^3\theta ,\alpha {\sin }^3\theta )$ and perpendicular to the line $x\sec \theta +ycosec\theta =\alpha$
Slope of line $x\sec \theta +ycosec\theta =\alpha$ is
$m=-\tan \theta$
$\Rightarrow \text{Slope of the required straight line is}\cot \theta$
$\Rightarrow y=\cot \theta x+c$
It passes through $(a{\cos }^3\theta ,\alpha {\sin }^3\theta )$
$c=\alpha {\sin }^3\theta -\alpha {\cos }^3\theta \cot \theta$
$c=-\frac{\alpha \cos 2\theta }{\sin \theta }$
$y=\cot \theta x-\frac{\alpha \cos 2\theta }{\sin \theta }$
$\therefore x\cos \theta -y\sin \theta =\alpha \cos 2\theta$