Explanation:
D Given,
\(y^2=4 a x\)
Any point on parabola (at \(\left.{ }^2, 2 a t\right)\)
Chord is passed through vertex \((0,0)\)
Midpoint may be \(\left(\frac{\mathrm{at}^2}{2}\right.\), at \()=(\mathrm{h}, \mathrm{k})\)
Where, \(\mathrm{h}=\frac{\mathrm{at}^2}{2}, \mathrm{k}=\mathrm{at}\)
\(2 \mathrm{~h}=\mathrm{a}\left(\frac{\mathrm{k}}{\mathrm{a}}\right)^2\)
\(2 \mathrm{ah}=\mathrm{k}^2\)
Replacing \(\mathrm{h}\) by \(\mathrm{x}\) and \(\mathrm{k}\) by \(\mathrm{y}\), we get-
\(2 a x=y^2\)
\(y^2=4\left(\frac{a}{2}\right) x\)Thus, directrix is, \(x=\frac{-a}{2}\)