The normal at a point $P$ on the ellipse $x^{2}+4y^{2}=16$ meets the $x$-axis at $Q$. If $M$ is the mid point of the line segment $PQ$, then the locus of $M$ intersects the latus rectums of the given ellipse at the points\\
(A) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{2}{7}\right)$\\
(B) $\left(\pm\frac{3\sqrt{5}}{2},\pm\frac{\sqrt{19}}{4}\right)$\\
(C) $\left(\pm2\sqrt{3},\pm\frac{1}{7}\right)$\\
(D) $\left(\pm2\sqrt{3},\pm\frac{4\sqrt{3}}{7}\right)$