Let Px1,y1 and Qx2,y2,y1<0,y2<0 , be the endpoints of the latus rectum of the ellipse x2+4y2=4. The
equations of parabolas with latusrectum PQ are
x2+23y=3+3
x2−23y=3+3
x2+23y=3−3
x2−23y=3−3
The given ellipse is
x24+y21=1b2=a21−e2or e=32
Hence, the endpoints P and Q of the latus rectum are given by
P≡3,-12and Q≡−3,−12
The coordinates of the midpoint of PQ are
R≡0,−12
Length of latus rectum, PQ=23
Hence, two parabolas are possible whose vertices are
S0,32-12 and T0,−32−12
The equation of the parabolas are
x2=23y+32+12and x2=−23y−32+12or x2−23y=3+3or x2+23y=3−3