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Options

A. `x^(2)+2sqrt(3)y=3+sqrt(3)`
B. `x^(2)-2sqrt(3)y=3+sqrt(3)`
C. `x^(2)+2sqrt(3)y=3-sqrt(3)`
D. `x^(2)-2sqrt(3)y=3-sqrt(3)`

Explanation

The given ellipse is
`(x^(2))/(4)+(y^(2))/(1)=1`
`b^(2)=a^(2)(1-e^(2))`
`or e=(sqrt(3))/(2)`

Hence, the endpoints P and Q of the latus rectum are given by
`P-=(sqrt(3),-(1)/(2))`
and `Q=-(-sqrt(3),-(1)/(3)) " " ("Given" y_(1),y_(2)lt0)`
The midpoint of PQ is
`R=-(0,(1)/(2))`
Hence two parabolas are possible are whose vertices are
`S(0,+(sqrt(3))/(2)-(1)/(2)) and T, (0,-(sqrt(3))/(2)-(1)/(2))`
the equations of the parabolas are
`x^(2)=2sqrt(3)(y+(sqrt(3))/(2)+(1)/(3))`
and `x^(2)= -2sqrt(3)(y-(sqrt(3))/(2)+(1)/(2))`
or `x^(2)-2sqrt(3)y=3+sqrt(3)`
and `x^(2)+2sqrt(3)=3-sqrt(3)`