Question

A variable circle is drawn to touch the $x$-axis at the origin. The locus of the pole of the straight line $\ell x+my+n=0$ w.r.t. the variable circle has the equation -

• A. $x(my-n)-\ell y^2=0$
• B. $x(my+n)-\ell y^2=0$
• C. $x(my-n)+\ell y^2=0$
• D. none

Answer 1

Answer: (A)

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