Question

lf the polar w.r.t the circle $x^2+y^2=r^2$ touches the parabola $y^2=4ax$, then locus of the pole is

• A. $y^2=\frac{-r^2}{a}x$
• B. $x^2=\frac{-r^2}{a}y$
• C. $y^2=\frac{r^2}{a}x$
• D. $x^2=\frac{r^2}{a}y$

Answer 1

The correct option is A $y^2=\frac{-r^2}{a}x$

Let $(h,k)$ be the pole

The equation of polar of the circle is $hx+ky=r^2$

$\Rightarrow y=\frac{-hx}{k}+\frac{r^2}{k}$

Since, it is tangent to parabola $y^2=4ax$

Then $c=\frac{a}{m}$

$\Rightarrow \frac{r^2}{k}=\frac{ak}{-h}$

Therefore, locus is $y^2=\frac{-r^{2}x}{a}$