Find the eccentricity of an ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` whose latus rectum is half of its major axis.

Options

A. `(1)/(sqrt(2))`
B. `sqrt((2)/(3))`
C. `(sqrt(3))/(2)`
D. None of these

Answer

A

Explanation

Let `a gt b`
Then the latus rectum of the ellipse is `(2b^(2))/(a)` and the semi-major axis is a. Given
`(2b^(2))/(a)=arArr2b^(2)=a^(2)`
Also for the ellipse, `b^(2)=a^(2)(1-e^(2))`
`rArr 2a^(2)(1-e^(2))=a^(2)`
`rArr e = (1)/(sqrt(2))`.