Question

Find the normal to the curve $x=a(1+cos\theta ),y=a\sin \theta$ at $\theta .$ Prove that it always passes through a fixed point and find that fixed point.

Answer 1

$x=a(1+cos\theta )\Rightarrow \frac{dx}{d\theta }=-asin\theta y=asin\theta \Rightarrow \frac{dy}{d\theta }=acos\theta \therefore \frac{dy}{d\theta }=\frac{\left(\frac{dy}{d\theta }\right)}{\left(\frac{dx}{d\theta }\right)}=\frac{acos\theta }{-asin\theta }=-cot\theta$

$\therefore$ Slope of normal $=-\frac{1}{\left(\frac{dy}{d\theta }\right)}=-\frac{1}{-cot\theta }=tan\theta$

The equation of normal $=(y-asin\theta )=tan\theta (x-a-acos\theta )=(y-asin\theta )=xtan\theta -atan\theta -asin\theta =y=xtan\theta -atan\theta =tan\theta (x-a)$

$\therefore y=tan\theta (x-a)$

Comparing it with $(y-y_1)=m(x-x_1)$, we find that the normal of the curve passes through fixed point. $(a,0)$