If two distinct chords drawn from the point ( p , q ) on the circle x 2 + y 2 = p x + q y (where p q ≠ 0 ) are bisected by the X -axis, then

Step-by-step solution

Let ( t ,   m ) be the other end of the chord drawn from the point ( p ,   q ) on the circle x 2 + y 2 = p x + q y Their mid-point is t + p 2 , m + q 2 since, mid-point lies on X -axis i.e. y = 0 m + q = 0 . . . ( i ) Also, ( t ,   m ) lies on the circle. ∴   t 2 + m 2 - p t - q m = 0 . . . ( ii ) From Eqs. ( i ) and ( ii ) , we get t 2 - p t + 2 q 2 = 0 Which is quadratic in t such that, Discriminant > 0 ⇒   p 2 - 8 q 2 > 0 ⇒ p 2 > 8 q 2