Question

The difference between the lengths 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 $2a{e}^2$

Length of the latus rectum $=\frac{2b^2}{a}$
and $e=\sqrt{1-\frac{b^2}{a^2}}$
$a^2{e}^2=a^2-b^2$
$\Rightarrow b^2=a^2-a^2{e}^2$
$\Rightarrow b^2=a^2(1-e^2)$
$\therefore$ Length of the latus rectum
$=\frac{2a^2(1-e^2)}{a}=2a(1-e^2)$
Length of the major axis=2a
Difference between length of latus rectum and length of major axis =$2a-2a(1-e^2)$
$=2a{e}^2$