Question

The normal to the curve $x=a(1+\cos \theta ),y=a\sin \theta$ at any point $\theta$ always passes through the fixed point:

• A. $(a,0)$
• B. $(0,a)$
• C. $(0,0)$
• D. $(a,a)$

Answer 1

The correct option is A $(a,0)$

$x=a(1+\cos \theta ),y=a\sin \theta$
Differentiating w.r.t $\theta$, we get
$\frac{dx}{d\theta }=a(-\sin \theta ),\frac{dy}{d\theta }=a\cos \theta$
$\therefore \frac{dy}{dx}=-\frac{\cos \theta }{\sin \theta }$
Therefore equation of normal at $(a(1+\cos \theta ),\sin \theta )$ is
$(y-a\sin \theta )=\frac{\sin \theta }{\cos \theta }(x-a(1+\cos \theta ))$
It is clear that in the given options normal passes through the point $(a,0)$