Question

The locus of the midpoint of the line segment joining the focus to a moving point on the parabola y² = 4ax is another parabola with the directrix

• A. x = -a
• B. x = -a/2
• C. x =0
• D. x = a/2

Answer 1

Answer: (C)

x =0

Answer 2

α = $\frac{at^{2} + \alpha}{2}\mspace{6mu},\mspace{6mu}\beta = \frac{2at + 0}{2}\mspace{6mu} \Rightarrow \mspace{6mu} 2\alpha = at^{2} + a$, at = β

∴ The locus of y² = $\frac{4a}{2}\left( x - \frac{a}{2} \right) = 4b(x - b),\mspace{6mu}\left( b = \frac{a}{2} \right)$

∴ Directrix is (x – b) + b = 0 or x =0.