The eccentricity of the ellipse `x^(2)+4y^(2)+8y-2x+1=0`, is

Options

A. `(sqrt(3))/(2)`
B. `(sqrt(5))/(2)`
C. `(1)/(2)`
D. `(1)/(4)`

Answer

A

Explanation

the equation of the ellipse is
`x^(2)+4y^(2)+8y-2x+1=0`,
`implies (x^(2)-2x+1)+4(y^(2)+2y+1)=4`
`implies (x-1)^(2)/(2^(2))+((y-1)^(2))/(1^(2))=1`
this is of the form `((x-h)^(2))/(a^(2))+((y-k)^(2))/(b^(2))=1.`
on compairing , we get a=2 and b=1
Let e be the eccentricity of the ellipse , then
`e=sqrt(1-(b^(2))/(a^(2)))implies e= sqrt(1-(1)/(4))=(sqrt(3))/(2)`