If two distinct chords, drawn from the point (p, q) on the circle x2 + y2 = px + qy (where pq &#8800

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If two distinct chords, drawn from the point (p, q) on the circle x2 + y2 = px + qy (where pq &#8800; 0) are bisected by the x-axis, then

Accepted Answer

Note, In solving a line and a circle there oftengenerate a quadratic equation and further we have to apply condition of Discriminant so question convert from coordinate to quadratic equation.
Form equation of circle it is clear that circle passes through origin Let AB is chord of the circle
A ≡ (p, q).C is mid point and coordinate of C is (h, 0)
Then coordinates of B are (- p + 2h, -q) and B lies on the circle x² + y² = px + qy, we have
(- p + 2h)² + (-q)² = p(- p + 2h) + q(-q)
⇒ p² + 4h² - 4 ph + q² = - p² + 2ph - q²
⇒ 2p² + 2p² - 6 ph + 4h² = 0
⇒ 2h² - 3ph + p² + q² = 0 ........(i)
There are given two distinct chords which are bisected at x-axis then, there will be two distinct values of h satisfying Eq. (i).
So, discriminant of this quadratic equation must be > 0.
⇒ D > 0
⇒ (-3p)² - 4.2(p² + q²) > 0
⇒ 9p² - 8p² - 8q² > 0
⇒ p² - 8q² > 0
⇒ p² > 8q²

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