Question

The equations of the tangent and normal at point $ (3,\,\,-2) $ of ellipse $ 4{{x}^{2}}+9{{y}^{2}}=36 $ are

• A. $ \frac{x}{3}-\frac{y}{2}=1,\,\,\frac{x}{2}+\frac{y}{3}=\frac{5}{6} $
• B. $ \frac{x}{3}+\frac{y}{2}=1,\,\,\frac{x}{2}-\frac{y}{3}=\frac{5}{6} $
• C. $ \frac{x}{2}+\frac{y}{3}=1,\,\,\frac{x}{3}-\frac{y}{2}=\frac{5}{6} $
• D. None of the above

Answer 1

Answer: (A)

Given equation of ellipse is $ 4{{x}^{2}}+9{{y}^{2}}=36 $ or $ \frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}=1 $
Tangent at point $ (3,-2) $ is $ \frac{(3)x}{9}+\frac{(-2)y}{4}= $
or $ \frac{x}{3}-\frac{y}{2}=1 $
$ \therefore $ Normal is $ \frac{x}{2}+\frac{y}{3}=k $
and it passes through point $ (3,-2) $ .
Then, $ \frac{3}{2}-\frac{2}{3}=k\Rightarrow k=\frac{5}{6} $
Hence, normal is $ \frac{x}{2}+\frac{y}{3}=\frac{5}{6} $
and tangent is $ \frac{x}{3}-\frac{y}{2}=1 $ .