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. $x(m y-n)-\ell y^2=0$
• B. $x(m y+n)-\ell y^2=0$
• C. $x(m y-n)+\ell y^2=0$
• 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)
$\frac{ h }{\ell}=\frac{ k +\frac{\lambda}{2}}{ m }=\frac{\lambda k }{2 n }$
$\Rightarrow mh =\ell k +\frac{\ell \lambda}{2} \text { and } nh =\frac{\ell \lambda k }{2}$
$\Rightarrow mh -\ell k +\frac{ nh }{ k }$
$ \Rightarrow mhk -\ell k ^2+ nh $
$\therefore x ( my - n )-\ell y ^2=0$