Question

The difference between the lengths of the major axis and the latus-rectum of an ellipse is

Options

• ae

• 2ae

• ae²

• 2ae²

Answer 1

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