Question

If the polar of a point w.r.t. circle $x^2+y^2=r^2$ touches the parabola $y^2=4ax$, the 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

Answer: (A)

Polar of a point $\left(x_1,y_1\right)$w.r.t.
$x^2+y^2=r^2$ is $x{x}_1+y{y}_1=r^2$
i.e., $y{y}_1=-x{x}_1+r^2$
$\Rightarrow y=-\frac{x_1}{y_1}x+\frac{r^2}{y_1}$
$\Rightarrow y=mx+c$, where $m=-\frac{x_1}{y_1};c=\frac{r^2}{y_1}$
This touches the parabola $y^2=4ax$ if
$c=\frac{a}{m}$
$\Rightarrow \frac{r^2}{y_1}=\frac{a}{-x_1/y_1}=-\frac{a{y}_1}{x_1}$
$\therefore$ reqd. locus of pole $\left(x_1,y_1\right)$ is
$\frac{r^2}{y}=-\frac{ay}{x}$
i.e., $y^2=-\frac{r^2}{a}x$