# 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 : Option: 1 $x=\frac{a}{2}$ Option: 2 $x=0$ Option: 3 $x=2$   Option: 4 $x=-\frac{a}{2}$

$\mathrm{h}=\frac{\mathrm{at}^{2}+\mathrm{a}}{2}, \mathrm{k}=\frac{2 \mathrm{at}+0}{2} \\ \\\Rightarrow \mathrm{t}^{2}=\frac{2 \mathrm{~h}-\mathrm{a}}{\mathrm{a}} \text { and } \mathrm{t}=\frac{\mathrm{k}}{\mathrm{a}}$

\begin{aligned} &\Rightarrow \frac{\mathrm{k}^{2}}{\mathrm{a}^{2}}=\frac{2 \mathrm{~h}-\mathrm{a}}{\mathrm{a}}\\ &\Rightarrow \text { Locus of }(\mathrm{h}, \mathrm{k}) \text { is } \mathrm{y}^{2}=\mathrm{a}(2 \mathrm{x}-\mathrm{a})\\ &\Rightarrow y^{2}=2 a\left(x-\frac{a}{2}\right) \end{aligned}

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

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