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

$\to x^2-px+2q^2=0-\left(1\right)$

which gives two values of x and hence the coordinates of two points Qand 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 (1) are real distinct.

the discriminant p

$p^2-8q>0\to p^2>8q^2$