Question

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

• A. $p^2=q^2$
• B. $p^2=8q^2$
• C. $p^2<8q^2$
• D. $p^2>8q^2$

Answer 1

The correct option is D $p^2>8q^2$

Let $PQ$ be a chord of the given circle passing through $P(p,q)$ and the coordinates of $Q$ be $(x.y)$ 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 $x^2+q^2-px+q^2=0$ so $x^2+q^2-px+q^2=0$ $\Rightarrow x^2-px+2q^2=\mathrm{0...}\left(i\right)$
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.
$\Rightarrow$ the discriminant $p^2-8q^2>0\Rightarrow p^2>8q^2$