Question

For the horizontal ellipse, the difference between the length of the major axis and the latus rectum of an ellipse is

• A. $ae$
• B. $2ae$
• C. $a{e}^2$
• D. $2a{e}^2$

Answer 1

The correct option is D $2a{e}^2$

Let the equation of ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$
Eccentricity $e=\sqrt{1-\frac{b^2}{a^2}}$
$\Rightarrow a^2{e}^2=a^2-b^2$
$\Rightarrow b^2=a^2(1-e^2)$

$\therefore$ Length of the latus rectum $=\frac{2b^2}{a}=2a(1-e^2)$
Length of the major axis $=2a$
$\therefore$ required distance is $=2a-2a(1-e^2)$
$=2a{e}^2$