Question

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

• A. $x = a$
• B. $x = -\frac{a}{2}$
• C. $x = 0$
• D. $x = \frac{a}{2}$

Hint:

The locus of the midpoint between the focus and any point on a parabola is another parabola with half the latus rectum and a shifted vertex.

Answer 1

The Correct Option is C

Step 1: Focus $S(a, 0)$. Moving point $P(at^2, 2at)$.
Step 2: Mid-point $(h, k) = \left(\frac{at^2+a}{2}, \frac{2at+0}{2}\right)$.
Step 3: $k = at \Rightarrow t = k/a$.
Step 4: $h = \frac{a(k/a)^2 + a}{2} \Rightarrow 2h = \frac{k^2}{a} + a \Rightarrow k^2 = 2a(h - a/2)$.
Step 5: Locus: $y^2 = 2a(x - a/2)$.
Step 6: For $Y^2 = 4AX$, directrix is $X = -A$. Here $4A = 2a \Rightarrow A = a/2$.
Step 7: $x - a/2 = -a/2 \Rightarrow x = 0$.