A) \[\frac{\pi }{6}\]
B) \[\frac{\pi }{4}\]
C) \[\frac{\pi }{3}\]
D) \[\frac{\pi }{2}\]
Correct Answer: D
Solution :
[d] Differentiating \[{{y}^{3}}-{{x}^{2}}y+5y-2x=0\]w.r.t. x, we get \[3{{y}^{2}}y'-2xy-{{x}^{2}}y'+5y'-2=0\] Or \[y'=\frac{2xy+2}{3{{y}^{2}}-{{x}^{2}}+5}\]or \[y{{'}_{(0,0)}}=\frac{2}{5}\] Differentiating \[{{x}^{4}}-{{x}^{3}}{{y}^{2}}+5x+2y=0\] w.r.t. x, we get \[4{{x}^{2}}-3{{x}^{2}}{{y}^{2}}-2{{x}^{3}}yy'+5+2y'=0\] Or \[y'=\frac{3{{x}^{2}}{{y}^{2}}-4{{x}^{3}}-5}{2-2{{x}^{3}}y}\]or \[y{{'}_{(0,0)}}=-\frac{5}{2}.\] Thus, both the curves intersect at right angle.
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