Ellipse

We have
x2+4y2 = 4

x2

4

+

y2

1

= 1
and
b2 = a2(1−e2)
1 = (4(1−e2)
⇒ e =

√3

2

Therefore, the points P and Q are
P(√3,−

1

2

) ; Q (−√3,−

1

2

)
Since P and Q are the endpoints of latus rectum of parabola
The midpoint of PQ is the focus of parabola. Hence,
focus = (0,−

1

2

)
Hence, the length of latus rectum = 4a. Therefore,
4a = 2√3 ⇒ a =

√3

2

Hence, the two parabolas are possible whose vertices are
(0,−

√3

2

−

1

2

) and (0,

√3

2

−

1

2

).
Hence, the equation of the parabolas are given as follows:
x2 = 2√3[y−(

√3

2

−

1

2

)]
and x2 = 2√3[y−(−

√3

2

−

1

2

)]