Question

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

• A. $x+ey+e^{3}a=0$
• B. $x-ey-e^{3}a=0$
• C. $x-ey-e^{2}a=0$
• D. None of these.

Answer 1

Answer: (B)

End of the Latus Rectum is $L\left(ae,\frac{b^2}{a}\right)$
Normal at $\left(x_1,y_1\right)$ is $\frac{x-x_1}{x_1/a^2}=\frac{y-y_1}{y_1/b^2}$
$\therefore$ Normal at $L$ is $\frac{x-e}{ae/a^2}$=$\frac{y-\frac{b^2}{a}}{\frac{\frac{b^2}{a}}{b^2}}$
$\Rightarrow \frac{a\left(x-ae\right)}{e}=ay-b^2$
$=ay-a^2\left(1-e^2\right)$
$\Rightarrow \frac{x-ae}{e}=y-a\left(1-e^2\right)$
$\Rightarrow x-ae=ey-ae+a{e}^3$
$\Rightarrow x=ey+a{e}^3$
$\Rightarrow x-ey-e^{3}a=0$