Question:
If two distinct chords from (p,q) on the circle
(x^2 + y^2) = (px+qy) are bisected by x-axis.Then a)p^2 = q^2 b)p^2 = 8q^2 c)p^2 > 8q^2 d)p^2 

Solution:

Since the x axis bisects the chord, so if (a,b) is the other end of chord.

Then ((a+p)/2,(b+q)/2) lies on the x axis. Hence b+q=0, b=-q.

So the pair of distinct chords are passing through (p,q) and (a,-q).

Putting (a,-q) in circle's equation we get a quadratic equation
(a^2 - pa + 2q^2) = 0 For pair of distinct roots:
D > 0 That is
(p^2 - 8q^2) > 0 p^2 > 8q^2!!