Given equation of an ellipse is
$x^{2}+4 y^{2}=4$
$\Rightarrow \frac{x^{2}}{4}+\frac{y^{2}}{1}=1$
$\therefore $ Coordinate of positive end of minor axis is $B(0,1)$. Let mid-point of the chord $B P$ is $M(h, k)$
Then, $\quad(h, k)=\left(\frac{0+x}{2}, \frac{1+y}{2}\right)$
$\Rightarrow h=\frac{x}{2} \quad \Rightarrow x=2 h$
and $k=\frac{1+y}{2} \Rightarrow y=2 k-1$
$\therefore P(x, y) \equiv\{2 h,(2 k-1)\}$
Since, the point ' $P$ ' lies cllipes so form Eq. (i), we get
$(2 \,h)_{1}^{2}+4(2 \,k-1)^{2}=4$
$\Rightarrow 4 \,h^{2}+4(2 k-1)^{2}=4$
$\Rightarrow h^{2}+4 \,k^{2}+1-4 \,k=1$
$\Rightarrow h^{2}+4 \,k^{2}-4 \,k=0$
Thus, required locus is an ellipse whose equation is $x^{2}+4\, y^{2}-4\,y=0$
$\Rightarrow \frac{(x-0)^{2}}{1}+\frac{\left(y-\frac{1}{2}\right)^{2}}{\left(\frac{1}{4}\right)}=1$
whose centre $\left(0, \frac{1}{2}\right)$ and major and minor axis $\frac{1}{2}$
