The locus of the mid-point of the line segment joining the focus of the parabola
Question:

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 :

 

  1. $x=-\frac{a}{2}$

  2. $x=\frac{a}{2}$

  3. $x=0$

  4. $\mathrm{X}=\mathrm{a}$


Correct Option: 3

Solution:

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

$\Rightarrow \quad \mathrm{t}^{2}=\frac{2 \mathrm{~h}-\mathrm{a}}{\mathrm{a}}$ 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 $x-\frac{a}{2}=-\frac{a}{2} \Rightarrow x=0$

 

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