Question

The locus of the midpoint of the line joining focus and any point on the parabola ${\mathrm{y}}^2=4\mathrm{ax}$ is a parabola with the equation of directrix as

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

Answer 1

The correct option is C

$\mathrm{x}=0$

Explanation of the correct option.

Compute the directrix:

Let the point $\mathrm{P}(\mathrm{h},\mathrm{k})$ be the midpoint of the line segment joining the focus $(\mathrm{a},0)$and a general point $\mathrm{Q}(\mathrm{x},\mathrm{y})$ on the parabola.

Therefore,

$$\begin{gathered} \mathrm{h}=\frac{[\mathrm{x}+\mathrm{a}]}{2} \\ \Rightarrow \mathrm{x}=2\mathrm{h}-\mathrm{a} \end{gathered}$$

$$\begin{gathered} \mathrm{k}=\frac{\mathrm{y}}{2} \\ \Rightarrow y=2k \end{gathered}$$

Substitute the values of $\mathrm{x}$ and $\mathrm{y}$ in ${\mathrm{y}}^2=4\mathrm{ax}$,

$$\begin{gathered} 4{\mathrm{k}}^2=4\mathrm{a}(2\mathrm{h}–\mathrm{a}) \\ \Rightarrow 4{\mathrm{k}}^2=8\mathrm{ah}-4{\mathrm{a}}^2 \\ \Rightarrow {\mathrm{k}}^2=2\mathrm{ah}-{\mathrm{a}}^2 \end{gathered}$$

The locus of $\mathrm{P}(\mathrm{h},\mathrm{k})$is

${\mathrm{y}}^2=2\mathrm{ax}–{\mathrm{a}}^2$.

$\Rightarrow {\mathrm{y}}^2=2\mathrm{a}\left[\mathrm{x}-\frac{\mathrm{a}}{2}\right]$

Therefore, Its directrix is

$$\begin{gathered} x-\frac{\mathrm{a}}{2}=–\frac{\mathrm{a}}{2} \\ \Rightarrow x=0 \end{gathered}$$.

Hence option $\mathrm{C}$ is the correct answer.