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Through the vertex O of a parabola y2 = 4x chords OP and OQ are drawn at right angles to one another. Show that for all position of P, PQ cuts the axis of the parabola at a fixed point. Also find the locus of the middle point of PQ. 

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The given parabola is y2 = 4x   .....(1)

This line cuts the x-axis (the axis of the parabola) at (4, 0) which is a fixed point for all positions of P.

Let R(α,β) be the middle point of PQ. Then α = t12 + t22  .....(3)

and β = t1 + t2

From (4),β2 = t1 2 + t2 2 + 2t1 t2 = 2α – 8 [From (2) and (3)] Hence Locus of R (αβ) is y2 = 2x – 8

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