- The locus of the mid-points of the line joining the focus and point on the parabola y²=4 a x is a parabola with the equation of directrix as

Step-by-step solution

Let the mid-point of any point P (x₁, y₁ ) on parabola and focus F(a, 0) be Q(h, k) . Then, (h, k)= ( x₁+a 2 , y₁+0 2 ) aligned & h= x₁+a 2 and k= y₁ 2 & x₁=2 h-a and y₁=2 k aligned Since, P (x₁, y₁ ) lies on parabola y²=4 a x , then array r (2 k)²=4 a(2 h-a) 4 k²=4 a(2 h-a) k²=2 a (h- a 2 ) y²=2 a (x- a 2 ) array Equation of its directrix is aligned x- a 2 &= -a 2 x &=0 aligned