PQ is a chord of the circle $x^2+y^2-2x-8=0$ whose midpoint is (2, 2). The circle passing through P, Q and (1, 2) is

The equation of the chord of the circle $x^2+y^2-2x-8=0$ having (2, 2) as its mid-point is

$\begin{array}{l}2x+2y-(x+2)-8=4+4-4-8 \text{ [Using: }{S}^{\mathrm{\prime }}=T\text{ ] }\\ \Rightarrow x+2y-6=0\end{array}$

The equation of a circle passing through P and Q is

$x^2+y^2-2x-8+\lambda (x+2y-6)=0$                       ... (i)

It passes through (1, 2).

$\therefore 1+4-2-8+\lambda (1+4-6)=0\Rightarrow \lambda =-5$

Putting the value of $\lambda$ in (i), we obtain

$x^2+y^2-7x-10y+22=0$

as the equation of the required circle.