Question

Tangents $TP$ and $TQ$ are draw n from a point $T$ to the circle $x^2+y^2=a^2$. If the point $T$ lies on the line $px+qy=r$, find the locus of centre of the circum-circle of triangle $TPQ$.

Answer 1

Circle $TPQ$ is $S+\lambda P=0$ where $P$ is chord of contact of $T(h,k)$.
Circumcircle is $(x^2+y^2-a^2)+\lambda (hx+ky-a^2)$
It passes through $T(h,k)$
$\therefore (h^2+k^2-a^2)+\lambda (h^2+k^2-a^2)=0$
or $\lambda =-1$
Circle is $x^2+y^2-hx-ky=0$
Its centre $x=\frac{h}{2},y=\frac{k}{2}$. But $(h,k)$ lies on
$px+qy+r=0,\therefore ph+qk+r=0$.
Eliminating $h,k$ we get the locus as
$p.2x+q.2y+r=0$.