The locus of the mid-point of the line
Question:

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

1. (1) $x=a$

2. (2) $x=0$

3. (3) $x=-\frac{a}{2}$

4. (4) $x=\frac{a}{2}$

Correct Option: , 2

Solution:

$\mathrm{h}=\frac{\mathrm{at}^{2}+\mathrm{a}}{2}, \mathrm{k}=\frac{2 \mathrm{at}+0}{2}$

$\Rightarrow \mathrm{t}^{2}=\frac{2 \mathrm{~h}-\mathrm{a}}{\mathrm{a}}$ and $\mathrm{t}=\frac{\mathrm{k}}{\mathrm{a}}$

$\Rightarrow \frac{k^{2}}{a^{2}}=\frac{2 h-a}{a}$

$\Rightarrow$ Locus of $(h, k)$ is $y^{2}=a(2 x-a)$

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

Its directrix is $x-\frac{a}{2}=-\frac{a}{2} \Rightarrow x=0$