The eccentricity of an ellipse with centre at the orgin and axes along the coordinate axes , is 1/2 if one of the directrices is x=4, the equation of the ellipse is

Options

A. `4x^(2)+3y^(2)=1`
B. `3x^(2)+4y^(2)=12`
C. `4x^(2)+2y^(2)=12`
D. `3x^(2)+4y^(2)=1`

Answer

B

Explanation

Let the equation of the ellipe be `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`.
we have ,
`e=(1)/(2) and (a)/(e) =4 [:' x=(a)/(e) " is a directrix"]`
`implies a=2 `
`implies b^(2)=2^(2)(1-(1)/(4)) [:' b^(2)=a^(2) (1-e^(2))]`
`implies b^(2)=3`
hence , the equation of the ellipse is `(x^(2))/(4)+(Y^(2))/(3)=1 or 3x^(2)+4y^(2)=12`.