A) \[{{a}^{2}}\]
B) 1
C) \[-1\]
D) \[-{{a}^{2}}\]
E) \[\pi \]
Correct Answer: C
Solution :
\[x=a{{\cos }^{4}}\theta \]and\[y=a{{\sin }^{4}}\theta \] On differentiating w.r.t.\[\theta \]respectively \[\therefore \] \[\frac{dx}{d\theta }=-4a{{\cos }^{3}}\theta \sin \theta \] and \[\frac{dx}{d\theta }=4a{{\sin }^{3}}\theta \cos \theta \] \[\therefore \] \[\frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }=-\frac{4a{{\sin }^{3}}\theta \cos \theta }{4a{{\cos }^{3}}\theta \sin \theta }=-{{\tan }^{2}}\theta \] Hence, \[{{\left( \frac{dy}{dx} \right)}_{\theta =\frac{3\pi }{4}}}=-{{\left( \tan \frac{3\pi }{4} \right)}^{2}}=-1\]
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