If two distinct chords drawn from the point ( p, q ) on the circle...

Let $( t, m )$ be the other end of the chord drawn from the point $( p, q )$ on the circle

$x^{2} + y^{2} = px + qy$

Their mid-point is \left(\dfrac{t + p}{2}, \dfrac{m + q}{2}\right)$</p><p>since, mid-point lies on$X$-axis i.e.$y = 0$</p><p>$m + q = 0~ &. . . ( i )$</p><p>Also,$( t, m )$lies on the circle.</p><p>$∴ t^{2} + m^{2} - pt - qm = 0~ &. . . ( ii )$</p><p>From Eqs.$( i ) and ( ii )$, we get$t^{2} - pt + 2q^{2} = 0$</p><p>Which is quadratic in$t$such that, Discriminant$> 0$</p><p>$⇒ p^{2} - 8q^{2} > 0 ⇒ p^{2} > 8q^{2}$