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

Step-by-step solution

Let the coordinates of focus be S(a, 0) . Let any point on the parabola be P (at ², 2 at). Let the coordinates of mid point of P and S be (x₁, y₁ ) . aligned & & x ₁ &= a + at ² 2 , y ₁ & & at ² &=2 x ₁- a , y ₁ & a ( y ₁ a )² &=2 x ₁- a & y ₁² &=2 x ₁ a - a ² aligned Hence, the locus of the mid point is y ²=2 a ( x - a 2 ) Equation of directrix is x- a 2 =- a 2 x=0