The locus of the middle points of chords of a parabola which subtend a right angle at the vertex of the parabola, is

Step-by-step solution

α = a r 1 2 + a t 2 2 2 ;   β = 2 a t 1 + 2 a t 2 2 = a f 2 + t 2 Also, 2 a t 1 - 0 a t 1 2 - 0 × 2 a t 2 - 0 a t 2 2 - 0 = - 1 ⇒   t 1 t 2 = - 4 Now, 2 α a = t 1 2 + t 2 2 = t 1 + t 2 2 - 2 t 1 t 2 = β a 2 - 2 · - 4 So, the locus of ( α , β ) is 2 x a = y 2 a 2 + 8 which is a parabola.