Question

A circle touches a straight line $\ell x + my + n =0$ and cuts the circle $x ^2+ y ^2=9$ orthogonally. The locus of centres of such circles is -

• A. $(\ell x+m y+n)^2=\left(\ell^2+m^2\right)\left(x^2+y^2-9\right)$
• B. $(\ell x+m y-n)^2=\left(\ell^2+m^2\right)\left(x^2+y^2-9\right)$
• C. $(\ell x+m y+n)^2=\left(\ell^2+m^2\right)\left(x^2+y^2+9\right)$
• D. none of these

Answer 1

Answer: (A)

Let the equation of the circle is -
$x^2+y^2+2 g x+2 f y+c=0$.....(1)
which touches the line $\ell x + my + n =0$
$\therefore\left|\frac{-\ell g-m f+n}{\sqrt{\ell^2+m^2}}\right|=\sqrt{g^2+f^2-c}$.......(2)
and circle (1) is orthogonal to the circle $x^2+y^2=9$
$\therefore 0 \,\,\,\, g+0 \,\,\,\, f=c-9 $
$\Rightarrow c=9$..........(3)
from (2) & (3)
$\left|\frac{-\ell g-m f+n}{\sqrt{\ell^2+m^2}}\right|=\sqrt{g^2+f^2-9}$
$\therefore$ locus of $(-g,-f)$ is
$(\ell x + my + n )^2=\left( x ^2+ y ^2-9\right)\left(\ell^2+ m ^2\right)$