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Q.
A point on the ellipse $4 x^2+9 y^2=36$, where the normal is parallel to the line $4 x-2 y-5=0$, is
Conic Sections
Solution:
Let $P \left( x _1, y _1\right)$ be the point on curve $4 x ^2+9 y ^2=36$, where the normal is parallel to line $4 x -2 y$ SiellisC $=5$. $=5$.
$\left.\therefore \frac{ dy }{ dx }\right|_{ P \left( x _1, y _1\right)}=\left(\frac{-4 x _1}{9 y _1}\right)$
Now, $\frac{9 y _1}{4 x _1}=2 \Rightarrow 9 y _1=8 x _1$
Also, $ 4 x_1^2+9 y_1^2=36$
$\therefore$ On solving (1) and (2), we get
$ P \left( x _1=\frac{9}{5}, y _1=\frac{8}{5}\right) \text { or }\left( x _1=\frac{-9}{5}, y _1=\frac{-8}{5}\right)$