Question

Find the eccentricity of the ellipse if the length of the latus rectum $=$half of the major axis.

Answer 1

Solve for the the eccentricity of the ellipse

Let the equation of ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Length of the major axis is $a$, so the length of the latus rectum is $\frac{2b^2}{a}$

According to condition given in question we can write

$a=\frac{2b^2}{a}$

$\Rightarrow a^2=2b^2$

$\therefore$The eccentricity of an ellipse is given as

$e=\sqrt{\left(1-\frac{b^2}{a^2}\right)}$ ……..(1)

Substitute $a^2=2b^2$ in the equation (1)

$$\begin{gathered} e=\sqrt{\left(1-\frac{b^2}{2b^2}\right)} \\ =\sqrt{\left(1-\frac{1}{2}\right)} \end{gathered}$$ { using $a^2=2b^2$}

$=\frac{1}{\sqrt{2}}$

Hence the required eccentricity of ellipse is $\frac{1}{\sqrt{2}}$