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

Answer 1

The correct option is B

$\mathrm{x}=0$

Explanation of the correct option.

Compute the directrix:

Given : $y^2=4ax$

Let's plot the graph.

Let $\mathrm{M}\left(\mathrm{h},\mathrm{k}\right)$ is the midpoint of the line segment.

From the graph,

$\mathrm{h}=\frac{\left({\mathrm{at}}^2+\mathrm{a}\right)}{2}$

$\Rightarrow {\mathrm{t}}^2=\frac{\left(2\mathrm{h}-\mathrm{a}\right)}{\mathrm{a}}$…………….$\left(1\right)$

$\mathrm{k}=\frac{\left(2\mathrm{at}+0\right)}{2}$

$\Rightarrow$ $\mathrm{t}=\frac{\mathrm{k}}{\mathrm{a}}$

Substitute the value of $\mathrm{t}$ in equation $1$,

$$\begin{gathered} {\left(\frac{\mathrm{k}}{\mathrm{a}}\right)}^2=\frac{\left(2\mathrm{h}-\mathrm{a}\right)}{\mathrm{a}} \\ \Rightarrow {\mathrm{k}}^2=\mathrm{a}(2\mathrm{h}-\mathrm{a}) \end{gathered}$$

Locus of $\mathrm{M}\left(\mathrm{h},\mathrm{k}\right)$ is

$$\begin{gathered} {\mathrm{y}}^2=\mathrm{a}\left(2\mathrm{x}-\mathrm{a}\right) \\ \Rightarrow {\mathrm{y}}^2=2\mathrm{a}(\mathrm{x}-\frac{\mathrm{a}}{2}) \end{gathered}$$

Therefore its directrix is

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

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