Question

The mid-point of the chord $AB$ of the circle $x^2+y^2-6x-4y+3=0$ is the point $(1,1)$. Determine the co-ordinates of the point of intersection of tangents to the circle at its extremities.

Answer 1

If $(h,k)$ be the required point, then $AB$ is chord of contact of $(h,k)$ and its equation is also given by $T=S_1$ as $(1,1)$ is its mid-point:

$2x+y=3$ by $T=S_1$

$x.h+y.k-3(x+h)-2(y+k)+3=0$

or $x(h-3)+y(k-2)=3h+2k-3$ as $C.C$.

Comparing, we get

$\frac{h-3}{2}=\frac{k-2}{1}=\frac{3h+2k-3}{3}$

Above will give two equations which when solved give $h=-1,k=0$.