Question

If the latus rectum of an ellipse is equal to half of minor axis, then find its eccentricity.

Answer 1

Let the equation of an ellipse is `x^2/a^2 + y^2/b^2` = 1

Length of major axis = 2a

Length of minor axis = 2b

And the length of latus rectum = `(2b^2)/a`

We have `(2b^2)/a = (2b)/2`

⇒ b = `a/2`

Now b² = a²(1 – e²), where e is the eccentricity

⇒ b² = 4b²(1 – e²)

⇒ 1 = 4(1 – e²)

⇒ 1 – e² = `1/4`

⇒ e² = `1 - 1/4`

⇒ e² = `3/4`

∴ e = `+- sqrt(3)/2`

So, e = `sqrt(3)/2`    ......[∵ e is not (–)]

Hence, the required value of eccentricity is `sqrt(3)/2`.