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. $x + ey + e^3a = 0$
• B. $ x - ey - e^3a = 0 $
• C. $x - ey - e^2a = 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+ae^{3} $
$\Rightarrow x=ey +ae^{3} $
$ \Rightarrow x-ey-e^{3}a = 0$