Find the equation of a line which passes through 2 cos 3 θ , 2 sin 3 θ and is perpendicular to the…

Given line is

$x\cos \theta -y\sin \theta =2\cos 2\theta ....\left(1\right)$

$\therefore$Slope of line $\left(1\right)$ is $\frac{\cos \theta }{\sin \theta }$

Slope of line perpendicular to line $\left(1\right)$ is $\frac{-\sin \theta }{\cos \theta }$

[$\because$ If $m_1$& $m_2$are slope of perpendicular lines then $m_1.m_2=-1$]

Equation of line passing through $\left(2{\cos }^3\theta ,2{\sin }^3\theta \right)$& perpendicular to line $\left(1\right)$ is

$y-2{\sin }^3\theta =\frac{-\sin \theta }{\cos \theta }\left(x-2{\cos }^3\theta \right)$

$\Rightarrow y\cos \theta -2{\sin }^3\theta .\cos \theta =-x\sin \theta +2{\cos }^3\theta .\sin \theta$

$\Rightarrow x\sin \theta +y\cos \theta =2{\cos }^3\theta .\sin \theta +2{\sin }^3\theta .\cos \theta$

$\Rightarrow x\sin \theta +y\cos \theta =2\sin \theta .\cos \theta \left({\cos }^2\theta +{\sin }^2\theta \right)$

$\Rightarrow \frac{x\sin \theta }{\sin \theta .\cos \theta }+\frac{y\cos \theta }{\sin \theta \cos \theta }=2$

$\therefore x\sec \theta +y\mathrm{cosec}\theta =2$