Question

The normal at an end of a latus rectum of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through an end of the minor axis if

• A. $e^4+e^2=1$
• B. $e^3+e^2=1$
• C. $e^2+e=1$
• D. $e^3+e=1$

Answer 1

The correct option is A $e^4+e^2=1$

Given ellipse equation is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Let $P(ae,\frac{b^2}{a})$ be one end of latus rectum.
Slope of normal at $P(ae,\frac{b^2}{a})=\frac{1}{e}$
Equation of normal is
$y-\frac{b^2}{a}=\frac{1}{e}(x-ae)$
It passes through$\acute{B}(0,b)$ then
$b-\frac{b^2}{a}=-a$
$a^2-b^2=-ab$
$a^4{e}^4=a^2{b}^2$
$e^4+e^2=1$