The locus of the mid-point of the line segment joining the focus of the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ to a moving point of the parabola, is another parabola whose directrix is :
Correct Option: 3
$\mathrm{h}=\frac{\mathrm{at}^{2}+\mathrm{a}}{2}, \mathrm{k}=\frac{2 \mathrm{a} t+0}{2}$
$\Rightarrow \quad \mathrm{t}^{2}=\frac{2 \mathrm{~h}-\mathrm{a}}{\mathrm{a}}$ and $\mathrm{t}=\frac{\mathrm{k}}{\mathrm{a}}$
$\Rightarrow \quad \frac{\mathrm{k}^{2}}{\mathrm{a}^{2}}=\frac{2 \mathrm{~h}-\mathrm{a}}{\mathrm{a}}$
$\Rightarrow$ Locus of $(h, k)$ is $y^{2}=a(2 x-a)$
$\Rightarrow \quad \mathrm{y}^{2}=2 \mathrm{a}\left(\mathrm{x}-\frac{\mathrm{a}}{2}\right)$
Its directrix is $x-\frac{a}{2}=-\frac{a}{2} \Rightarrow x=0$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.