If a latus rectum of an ellipse subtends a right angle at the centre of the ellipse, then write the eccentricity of the ellipse.

Options

A. `e=frac(sqrt(10)-1)(9)`
B. `e=frac(sqrt(5)-1)(2)`
C. `e=frac(sqrt(9)+1)(16)`
D. `e=frac(sqrt(9)+1)(2)`

Answer

`e=frac(sqrt(5)-1)(2)`

Explanation

For ellipse `frac(x^(2))(a^(2))+frac(y^(2))(b^(2))=1`

length of latus-rectum is `frac(2 b^(2))(a)` and e is eccentricity of an ellipse.

Height `h` of right angle triangle divides hypotenous in two parts `p` and `q` such that

`h^(2)=p q`


In this right angle triangle formed by latus-rectum, `ae `(`x`co-ordinate of latus rectum ) is height and length `frac(2 b^(2))(a)` is hypotenous.

According to above condition,


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