Question

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

• A. $x(my-n)-l{y}^2=0$
• B. $x(my+n)-l{y}^2=0$
• C. $x(my-n)+l{y}^2=0$
• D. None

Answer 1

The correct option is A $x(my-n)-l{y}^2=0$

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
$hx+ky+\frac{\lambda }{2}(y+k)=0$
$hx+(k+\frac{\lambda }{2})y+\frac{\lambda k}{2}=\mathrm{0.......}\left(1\right)$
and the equation of given polar is
$lx+my+m=\mathrm{0.........}\left(2\right)$
comparing $\left(1\right)$ and $\left(2\right)$
$\frac{h}{l}=\frac{k+\frac{\lambda }{2}}{m}=\frac{\lambda k}{2n}$
$\Rightarrow$ $mh=lk+\frac{l\lambda }{2}$ and $nh=\frac{l\lambda k}{2}$
$\Rightarrow$ $mh=lk+\frac{nh}{k}$ $\Rightarrow$ $mhk=l{k}^2+nh$
$\therefore$ $x(my-n)-l{y}^2=0$