Question

Find the locus of the pole of the line $lx+my+n=0$ with respect to the circle which touches $y$-axis at the origin

• A. $y(x+n)=m{x}^2$
• B. $y(lx-n)=m{x}^2$
• C. $y(lx-n)=m{x}^2{x}^2$
• D. none of these

Answer 1

The correct option is D none of these

Equation of variable circle which touch the y-axis at origin is $x^2+y^2+\lambda x=0$
Let the pole of the above circle be $P(h,k)$
Equation of polar is
$hx+ky+\frac{\lambda }{2}(x+k)=0$
$ky+(k+\frac{\lambda }{2})x+\frac{\lambda k}{2}=\mathrm{0.......}\left(1\right)$
and the equation of given polar is
$lx+my+m=\mathrm{0.........}\left(2\right)$
comparing $\left(1\right)$ and $\left(2\right)$
$\frac{h}{m}=\frac{k+\frac{\lambda }{2}}{l}=\frac{\lambda k}{2n}$
$\Rightarrow$ $lh=mk+\frac{m\lambda }{2}$ and $nh=\frac{m\lambda k}{2}$
$\Rightarrow$ $lh=mk+\frac{nh}{k}$ $\Rightarrow$ $lhk=m{k}^2+nh$
$\therefore$ $y(my-n)=l{x}^2=0$