Question

The curves $ 4{{x}^{2}}+9{{y}^{2}}=72 $ and $ {{x}^{2}}-{{y}^{2}}=5 $ at (3, 2):

• A. touch each other
• B. cut orthogonally
• C. intersect at $ 45{}^\circ $
• D. intersect at $ 60{}^\circ $
• E. none of these

Answer 1

Answer: (B)

The given curves are $ 4{{x}^{2}}+9{{y}^{2}}=72 $ ...(i) and $ {{x}^{2}}-{{y}^{2}}=5 $ ?(ii) On differentiating Eq. (i) w. r. t. $ x, $ we get $ 8x+18y\frac{dy}{dx}=0 $ $ \frac{dy}{dx}=-\frac{4x}{9y} $ $ \therefore $ Slope of Eq.(i) $ ={{m}_{1}}={{\left( \frac{dy}{dx} \right)}_{(3,2)}}=-\frac{2}{3} $ On differenting Eq. (ii) w. r. t. $ x, $ we get $ 2x-2y\frac{dy}{dx}=0 $ $ \Rightarrow $ $ \frac{dy}{dx}=\frac{x}{y} $ $ \therefore $ Slope of Eq.(ii) $ ={{m}^{2}}={{\left( \frac{dy}{dx} \right)}_{(3,2)}}=\frac{3}{2} $ $ \therefore $ $ {{m}_{1}}{{m}_{2}}=\frac{-2}{3}\times \frac{3}{2}=-1 $ $ \therefore $ Both the curves cut orthogonally.