Explanation

(d)
Suppose $AB$ is a chord of the circle through $A(p,\ q)$ having $M(h,\ 0)$ as its mid point. The coordinates of$\ B$ are $( - p + 2h,\ - q)$
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As $B$ lies on the circle
$x^{2} + y^{2} = px + qy$, we have
$( - p + 2h)^{2} + ( - q)^{2} = p( - p + 2h) + q( - q)$
$\Rightarrow 2p^{2} + 2q^{2} - 6ph + 4h^{2} = 0$
$\Rightarrow 2h^{2} - 3ph + p^{2} + q^{2} = 0$ …(i)
As there are two distinct chords from $A(p,\ q)$ which are bisected on $x$-axis, there must be two distinct values of $h$ satisfying Eq. (i)
$D = 9p^{2} - (4)(2)\left( p^{2} + q^{2} \right) > 0$
$\Rightarrow \ \ p^{2} > 8q^{2}$