If the chord of contact of tangents from a point $P$ to a given circle passes through $Q,$ then the circle on $PQ$ as diameter

 

Let $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right)$ be the given points and $x^2+y^2=a^2$ be the circle. The chord of contact of tangents drawn from

$P\left(x_1,y_1\right)\text{ to }{x}^2+y^2=a^2$ is

$x{x}_1+y{y}_1=a^2$

If it passes through $Q\left(x_2,y_2\right)$ then 

$x_1{x}_2+y_1{y}_2=a^2$

The equation of the circle on $PQ$ as diameter is

$\begin{array}{l}\left(x-x_1\right)\left(x-x_2\right)+\left(y-y_1\right)\left(y-y_2\right)=0\\ \Rightarrow x^2+y^2-x\left(x_1+x_2\right)-y\left(y_1+y_2\right)+x_1{x}_2+y_1{y}_2=0\end{array}$

This circle will cut the given circle orthogonally, if

$0\left(x_1+x_2\right)+0\left(y_1+y_2\right)=a^2+x_1{x}_2+y_1{y}_2$ 

$\Rightarrow x_1{x}_2+y_1{y}_2+a^2=0$ which is true