A) cut at right angles
B) touch each other
C) cut at an angle\[\frac{\pi }{3}\]
D) cut at an angle\[\frac{\pi }{4}\]
Correct Answer: A
Solution :
Key Idea: The angle between the curves is the angle between the tangents to the curve. We have\[,\] \[{{x}^{3}}-3x{{y}^{2}}+2=0\] ... (i) and \[3{{x}^{2}}y-{{y}^{3}}-2=0\] ... (ii) On differentiating Eq. (i) and (ii) with respect to\[x\], we get \[{{\left( \frac{dy}{dx} \right)}_{{{c}_{1}}}}={{m}_{1}}\frac{{{x}^{2}}-{{y}^{2}}}{xy}\] and \[{{\left( \frac{dy}{dx} \right)}_{{{c}_{2}}}}={{m}_{2}}=\frac{-2xy}{{{x}^{2}}-{{y}^{2}}}\] Since \[{{m}_{1}}\times {{m}_{2}}=-1\] Hence, the two curves cut at right angles.
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