The tangent P T and the normal P N to the parabola y^2=4 a x at a point P on

Step-by-step solution

Equation of tangent and normal at point P (a t^2, 2 a t ) is t y=x=a t^2 and y=-t x+2 a t+a t^2 Let centroid of P T N is R(h, k) . h= a t^2+ (-a t^2 )+2 a+a t^2 3 and k= 2 a t 3 array rlrl & & 3 h & =2 a+a ( 3 k 2 a )^2 & & 3 h & =2 a+ 9 k^2 4 a & 9 k^2 & =4 a(3 h-2 a) array Locus of centroid is y^2= 4 a 3 (x- 2 a 3 ) Vertex ( 2 a 3 , 0 ) ; directrix x- 2 a 3 =- a 3 x= a 3 Latusrectum = 4 a 3 Focus ( a 3 + 2 a 3 , 0 ) i.e. (a, 0)