Question

If the length of latus rectum of an ellipse is $4$ units and the distance between a focus and its farthest vertex on the major axis is $6$ units, then its eccentricity

• A. $\frac{2}{3}$
• B. $\frac{1}{3}$
• C. $\frac{1}{5}$
• D. $\frac{1}{12}$

Answer 1

The correct option is A $\frac{2}{3}$

Given, Length of latus rectum $=4$

$\therefore \frac{2b^2}{a}=4$

$\therefore b^2=2a$ ...(i)

$a+ae=6$

$\Rightarrow a(1+e)=6$

$\Rightarrow a(1+\sqrt{1-\frac{b^2}{a^2}})=6$

$\Rightarrow a(1+\frac{\sqrt{a^2-2a}}{a})=6$

$\Rightarrow a+\sqrt{a^2-2a}=6$

$\Rightarrow a^2-2a=(6-a{)}^2$

$\Rightarrow a^2-2a=36+a^2-12a$

$\Rightarrow 12a-2a=36$

$\Rightarrow a=\frac{36}{10}$

$\therefore \frac{36}{10}(1+e)=6$

$\Rightarrow 1+e=\frac{10}{6}$

$\Rightarrow e=\frac{10}{6}-1$

$\Rightarrow e=\frac{4}{6}$

$\therefore e=\frac{2}{3}$