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 latus rectum PQ are

The given ellipse isx24+y21=1b2=a21−e2or e=32Hence, the endpoints P and Q of the latus rectum are given byP≡3,−12and Q≡−3,−12(Given y1, y2 < 0)The midpoint of PQ isR≡0,−12length of latus rectum, PQ=23Hence, two parabolas are possible whose vertices areS0,+32−12 and T0,−32−12the equations of the Parabolas are    x2=23y+32+12 and     x2=−23y−32+12 or     x2−23y=3+3 and     x2+23y=3−3