If two distinct chords, drawn from the point $(p,q)$ on the circle $x^2+y^2=px+qy,$ where $pq\ne 0,$ are bisected by the $x$-axis, then

$M(\alpha ,0)$ is on the $x$-axis. 
$M$ is the midpoint of $AB.$
$\therefore B\equiv (2\alpha -p,-q)$
which lies on the circle 
$x^2+y^2=px+qy$
$\therefore (2\alpha -p{)}^2+q^2=p(2\alpha -p)-q^2,$
     or $2{\alpha }^2-3p\alpha +p^2+q^2=0$
As these are two chords, there are two values of $\alpha .$
$\therefore$ Discriminant$>0.$
$\Rightarrow 9p^2-8\left(p^2+q^2\right)>0$ or $p^2>8q^2.$