# State whether the following statement is true or false and justify : Equation of the line passing through the point $(a \cos^3 \theta, a \sin^3 \theta)$ and perpendicular to the line $x \sec \theta + y \: cosec \theta = a$ is $x \cos \theta - y \sin \theta = a \sin 2 \theta$

Toolbox:
• Equation of a line passing through a point $(x_1, y_1)$ and having slope $m$ is $y-y_1=m(x-x_1)$
• If two lines are perpendicular then the product of their slope is -1.
Step 1 :
Equation of the given line is
$x \sec \theta + y \: cosec \theta = a$
Slope of this line is
$-\large\frac{\sec \theta}{cosec \theta } $$= -\large\frac{\sin \theta }{\cos \theta } Hence slope of the line perpendicular to the above line is \large\frac{\cos \theta}{\sin \theta} It passes through the point ( a \cos^3 \theta , a \sin^3 \theta ) \therefore Equation of the required line is (y- a \sin^3 \theta ) = \large\frac{\cos \theta }{\sin \theta}$$ (x - a \cos^3 \theta )$
$\Rightarrow \sin \theta (y- a \sin^3 \theta ) = \cos \theta (x - a \cos^3 \theta )$
$\Rightarrow y \sin \theta - a \sin^4 \theta = x \cos \theta - a \cos^4 \theta$
$\Rightarrow x \cos \theta - y \sin \theta = a \cos^4 \theta - a \sin^4 \theta$
$\Rightarrow x \cos \theta - y \sin \theta = a ( cos^2 \theta + \sin^2 \theta ) ( \cos^2 \theta - \sin^2 \theta )$
But $\cos^2 \theta + \sin^2 \theta = 1$ and $\cos^2 \theta - \sin^2 \theta = \cos 2\theta$
$\therefore x \cos \theta - y \sin \theta = a \cos 2 \theta$
Hence the statement is true.