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 $xx_{1} +yy_{1} = r^{2} $
i.e., $yy_{1} = -xx_{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{ay_{1}}{x_{1}} $
$ \therefore $ reqd. locus of pole $\left(x_{1}, y_{1}\right)$ is
$ \frac{r^{2}}{y} = -\frac{ay}{x}$
i.e., $y^{2} = -\frac{r^{2}}{a}x$