Let \(P\left(x_1, y_1\right)\) and \(Q\left(x_2, y_2\right), y_1 < 0, y_2 < 0\), be the end points of the latusrectum of the ellipse \(x^2+4 y^2=4\). The equations of parabolas with latusrectum \(P Q\) are

The equation \(x^2+4 y^2=4\) represents an ellipse with 2 and 1 as semi-major and semi-minor axes and eccentricity \(\frac{\sqrt{3}}{2}\). Thus, the ends of latusrect are \(\left(\sqrt{3}, \frac{1}{2}\right)\) and \(\left(\sqrt{3},-\frac{1}{2}\right),\left(-\sqrt{3},-\frac{1}{2}\right)\) and \(\left(\sqrt{3},-\frac{1}{2}\right)\).


According to the question, we consider only \(P\left(-\sqrt{3},-\frac{1}{2}\right)\) and \(Q\left(\sqrt{3},-\frac{1}{2}\right)\).
Now, \(\quad P Q=2 \sqrt{3}\)
Thus, the coordinates of the vertex of the parabolas are \(A\left(0, \frac{-1+\sqrt{3}}{2}\right)\) and \(A^{\prime}\left(0, \frac{-1-\sqrt{3}}{2}\right)\) and corresponding equations are and
\[
(x-0)^2=-4 \cdot \frac{\sqrt{3}}{2}\left(y+\frac{1-\sqrt{3}}{2}\right)
\]
i.e., \((x-0)^2=4 \cdot \frac{\sqrt{3}}{2}\left(y-\frac{-1-\sqrt{3}}{2}\right)\) and
\[
\begin{aligned}
& x^2+2 \sqrt{3} y=3-\sqrt{3} \\
& x^2-2 \sqrt{3} y=3+\sqrt{3}
\end{aligned}
\]