If a perpendicular drawn through the vertex O of the parabola y^2=4 a x to any of its tangent meets the tangent at N and the parabola at M , then O N O M=

Step-by-step solution

Let a point P (a t^2, 2 a t ) over the parabola y^2=4 a x . So, equation of tangent at point P is aligned & So, points N (- a t^2 1+t^2 , a t^3 1+t^2 ) & and M ( 4 a t^2 ,- 4 a t ) aligned So, aligned O N O M & = a^2 t^4 (1+t^2 )^2 + a^2 t^6 (1+t^2 )^2 16 a^2 t^4 + 16 a^2 t^2 & = 4 a^2 t^2 t (1+t^2 ) 1+t^2 1 t^2 +1 =4 a^2 . aligned