If two distinct chords, drawn from the point on the circle (where, ) are bisected by the X-axis, then
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: D
Solution
Let a chord from (p, q) be bisected at point (h, 0) on the X-axis. The midpoint condition for a chord of x² + y² = px + qy (rewritten as x² + y² - px - qy = 0) gives T = S₁:
hx + 0·y - p(x+h)/2 - q(y+0)/2 = h² - ph - 0
This line must pass through (p, q). Using the standard midpoint-chord relation: the chord with midpoint (h, 0) has slope = -(h - p/2)/(0 - q/2) = (2h - p)/q.
The slope of the line joining (p, q) to (h, 0) is (q - 0)/(p - h) = q/(p - h).
Setting these equal: (2h - p)/q = q/(p - h)
(2h - p)(p - h) = q²
-2h² + 3ph - p² = q²
2h² - 3ph + (p² + q²) = 0
For two distinct real values of h (two distinct chords), the discriminant must be positive:
9p² - 8(p² + q²) > 0
p² - 8q² > 0
p² > 8q²
hx + 0·y - p(x+h)/2 - q(y+0)/2 = h² - ph - 0
This line must pass through (p, q). Using the standard midpoint-chord relation: the chord with midpoint (h, 0) has slope = -(h - p/2)/(0 - q/2) = (2h - p)/q.
The slope of the line joining (p, q) to (h, 0) is (q - 0)/(p - h) = q/(p - h).
Setting these equal: (2h - p)/q = q/(p - h)
(2h - p)(p - h) = q²
-2h² + 3ph - p² = q²
2h² - 3ph + (p² + q²) = 0
For two distinct real values of h (two distinct chords), the discriminant must be positive:
9p² - 8(p² + q²) > 0
p² - 8q² > 0
p² > 8q²
[1999]