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

Answer 1

Answer: (A)

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