The locus of the mid-point of the line segment joining the focus of the parabola y 2 = 4 a x to a moving point of the parabola, is another parabola whose directrix is:

Step-by-step solution

h = a t 2 + a 2 ,   k = 2 a t + 0 2 ⇒    t 2 = 2   h - a a and t = k a ⇒    k 2 a 2 = 2   h - a a ⇒ Locus of h ,   k is y 2 = a 2 x - a ⇒    y 2 = 2 a x - a 2 Its directrix is x - a 2 = - a 2 ⇒ x = 0