Question

The equation of the normal to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the end of the latus rectum in the first quadrant is

• A. $x+ey-a{e}^3=0$
• B. $x-ey+a{e}^3=0$
• C. $x-ey-a{e}^3=0$
• D. None of these

Answer 1

Answer: (C)

$x-ey-a{e}^3=0$

The end of the latus rectum in the first quadrant is $(ae,\frac{b^2}{a})$

Equation of normal at $(ae,\frac{b^2}{a})$ is

$\frac{a^{2}x}{ae}-\frac{b^{2}y}{b^2/a}=a^2-b^2[\frac{a^{2}x}{x_1}-\frac{b^{2}y}{y_1}=a^2-b^2]$

$\Rightarrow \frac{a}{e}x-ay=a^2{e}^2[\because e^2=\frac{a^2-b^2}{a^2}]\Rightarrow x-ey-a{e}^3=0$