Explanation:
D Let, Q(h, k) be the mid-point of line joining the focus \(\mathrm{f}(\mathrm{a}, 0)\) and variable point \(\mathrm{p}\left(\mathrm{x}_0, \mathrm{y}_0\right)\)
\(\therefore \quad (\mathrm{h}, \mathrm{k})=\left(\frac{\mathrm{x}_0+\mathrm{a}}{2}, \frac{\mathrm{y}_0}{2}\right)\)
\(\mathrm{h}=\frac{\mathrm{x}_0+\mathrm{a}}{2} \text { and } \mathrm{k}=\frac{\mathrm{y}_0}{2}\)
\(\mathrm{x}_0=2 \mathrm{~h}-\mathrm{a} \text { and } \mathrm{y}_0=2 \mathrm{k}\)
Since, \(\mathrm{p}\left(\mathrm{x}_0, \mathrm{y}_0\right)\) lies on parabola \(\mathrm{y}^2=4 \mathrm{ax}\)
\(\therefore \quad \mathrm{y}_0^2=4 \mathrm{ax}_0\)
\((2 \mathrm{k})^2=4 \mathrm{a}(2 \mathrm{~h}-\mathrm{a})\)
\(4 \mathrm{k}^2=4 \mathrm{a}(2 \mathrm{~h}-\mathrm{a})\)
\(\mathrm{k}^2=2 \mathrm{a}\left(\mathrm{h}-\frac{\mathrm{a}}{2}\right)\)
Which is equation of parabola,
\(\therefore \quad y^2=2 a\left(x-\frac{a}{2}\right)\)
\(\therefore\) Equation of directrix is given by
\(x-\frac{a}{2}=\frac{-a}{2}\)
\(x=0\)