Mathematics - Circle Question with Solution | TestHub

Let PQ be a chord of the given circle passing through $\mathrm{P}(\mathrm{p},\mathrm{q})$ ad the coordinates of Q be $(\mathrm{x},\mathrm{y})$. Since $PQ$ is bisected by the $x$-axis, the mid-point of $PQ$ lies on the $x$-axis which gives $y=-q$

Now Q lies on the circle $x^2+y^{2}px-qy=0$

So $x^2+q^2-px+q^2=0$

$\Rightarrow x^2-px+2q^2=0$

Which gives two values of x and hence the coordinates of two points Q and R (say), so that the chords PQ and PR are bisected by x-axis. If the chords PQ and PR are distinct, the roots of (i) are real distinct.

⇒ the discriminant $p^2-8q^2>0\Rightarrow p^2>8q^2$