Question

Equation of the line passing through the point $(a{\cos }^3\theta ,a{\sin }^3\theta )$ and perpendicular to the line $x\sec \theta +ycosec\theta =a$ is $x\cos \theta -y\sin \theta =a\sin 2\theta$. State whether the statement is true or false. Justify

Answer 1

Given line : $x\sec \theta +ycosec\theta =a$

$\Rightarrow ycosec\theta =-x\sec \theta +a$

$\Rightarrow \frac{y}{\sin \theta }=-\frac{x}{\cos \theta }+a$

$\Rightarrow y=-\frac{\sin \theta }{\cos \theta }\left(x\right)+a\sin \theta$

Slope of this line is, $m=-\tan \theta$

Line passing through the point

$(a{\cos }^3\theta ,a{\sin }^3\theta )$ is given by:

$y-a{\sin }^3\theta =m_1(x-a{\cos }^3\theta )$ …(i)

As both lines are perpendicular, so

$m\times m_1=-1$

$\Rightarrow m_1=\frac{-1}{m}=\cot \theta$

Putting this in equation (i), we get

$y-a{\sin }^3\theta =\cot \theta (x-a{\cos }^3$

$\Rightarrow \sin \theta (y-a{\sin }^3\theta )=\cos \theta (x-a{\cos }^3\theta )$

$\Rightarrow x\cos \theta -y\sin \theta =a({\cos }^4\theta -{\sin }^4\theta )$

$\Rightarrow x\cos \theta -y\sin \theta$

$=a({\cos }^2\theta -{\sin }^2\theta )({\cos }^2\theta +{\sin }^2\theta )$

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

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

Hence, statement is false.