A) \[\sqrt[3]{\frac{y}{x}}\]
B) \[\sqrt[3]{\frac{x}{y}}\]
C) \[-\sqrt[3]{\frac{x}{y}}\]
D) \[-\sqrt[3]{\frac{y}{x}}\]
Correct Answer: D
Solution :
If \[x=a{{\cos }^{3}}\theta \]and\[y=a{{\sin }^{3}}\theta \] \[\Rightarrow \] \[\frac{dx}{d\theta }=3a{{\cos }^{2}}\theta (-sin\theta )\] \[\Rightarrow \] \[\frac{dy}{d\theta }=3asi{{n}^{2}}\theta (cos\theta )\] \[\Rightarrow \] \[\frac{dy}{dx}=\frac{dy}{d\theta }\times \frac{d\theta }{dx}\] \[=(3a{{\sin }^{2}}\theta .\cos \theta ).\frac{1}{-(3a\sin \theta .{{\cos }^{2}}\theta )}\] \[\frac{dy}{d\theta }=-\frac{\sin \theta }{\cos \theta },\frac{dy}{dx}=-\tan \theta \] \[\Rightarrow \] \[\frac{dy}{dx}=-\frac{{{(y-a)}^{-1/3}}}{{{(x/a)}^{1/3}}}\] \[=-{{\left[ \frac{y}{a}\times \frac{a}{x} \right]}^{1/3}}\] \[\frac{dy}{dx}=-{{\left( \frac{y}{x} \right)}^{-1/3}}=\sqrt[3]{\frac{y}{x}}\]
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