Question

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

Answer 1

Let the equation of the ellipse be

\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a> b\)

Given, Latus rectum LL′ subtends right angle at the centre. 

∴ ∠LOL′ = 90^° 

Since OF is the median of ΔOLL′ 

∴ ∠LOF = ∠L'OF = 45^°

In the right angles triangles OFL, we have

\(tan45^° = \frac{OF}{LF}\)

Since 0 < e < 1, so

\(e = \frac{-1+\sqrt5}{2}\)