Question

If tangents be drawn to the circle $x^2+y^2=12$ at its points of intersection with the circle $x^2+y^2-5x+3y-2=0$, find the co-ordinates of their point of intersection.

Answer 1

$S=x^2+y^2-12=0$
$S^{\prime }=x^2+y^2-5x+3y-2=0$.
Equation of the common chord is $S-S^{\prime }=0$.
or $5x-3y=10.....\left(1\right)$
If the tangents to the circle $x^2+y^2=12$ at the extremities of the chord $\left(1\right)$ intersect at the point $(h,k)$ then this chord is the chord of contact of the point $(h,k)$ w.r.t. the circle $x^2+y^2=12$ is
$hx+ky=12....\left(2\right)$
Comparing $\left(1\right)$ and $\left(2\right)$, we get
$\frac{h}{5}=\frac{k}{-3}=\frac{12}{10}$ or $(h,k)=(6,-\frac{18}{5})$.