A) touch each other
B) cut orthogonally
C) intersect at\[45{}^\circ \]
D) intersect at\[60{}^\circ \]
E) none of these
Correct Answer: B
Solution :
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.
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