Through the vertex O of a parabola y ²=4 x , chords OP and O Q are drawn at right angles to one another. The locus of the middle point of PQ is

Step-by-step solution

Given parabola is y ²=4 ...(1) x Let P ( t ₁², 2 t ₁ ) and Q ( t ₂², 2 t ₂ ) Slope of OP = 2 t ₁ t ₁² = 2 t ₁ and slope of OQ = 2 t ₂ Since OP OQ , 4 t ₁ t ₂ =-1 or t ₁ t ₂=-4 (2) Let R ( h , k ) be the middle point of PQ , then h = t ₁²+ t ₂² 2 (3) and k = t ₁+ t ₂ (4) From (4), k ²= t ₁²+ t ₂²+2 t ₁ t ₂=2 ~h -8 [ From (2) and (3)] Hence locus of R ( h , k ) is y ²=2 x -8