Q.262. Property of the locus of chords of parabola subtending a right angle at the origin.

Question 262.
The variable chord PQ on the parabola with equation y^2 = 4x substends a right angle at the origin O. By taking P as (t1^2,2t1) and Q as (t2^2,2t2), find a relation between t1 and t2 and hence show that PQ passes through a fixed point on the x-axis.

Answer 262.
Angle POQ = 90°
=> PO is perpendicular to QO
=> product of their slopes = -1
=> (2t1/t1^2) * (2t2/t2^2) = -1
=> t1 t2 = - 4
If PO passes through R(a, 0) on x-axis, we need to prove that a is independent of t1 and t2.
As P, R and Q are collinear slope of PR = slope of QR
=> (2t1) / (t1^2 - a) = 2t2 / (t2^2 - a)
=> t1t2^2 - at1 = t2t1^2 - at2
=> a (t2 - t1) = t1t2 (t1 - t2)
=> a = - t1t2 = 4
=> PQ passes through (4, 0), a fixed point on x-axis.

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