Question

If $x=aCo{s}^3\theta$ and $y=aSi{n}^3\theta$ then $\frac{dy}{dx}$=_________

• A. $\sqrt[-3]{\frac{x}{y}}$
• B. $\sqrt[-3]{\frac{y}{x}}$
• C. $\sqrt[3]{\frac{y}{x}}$
• D. $\sqrt[3]{\frac{x}{y}}$

Answer 1

Answer: (B)

If $x=a{\cos }^3\theta$ and $y=a{\sin }^3\theta$
$\Rightarrow \frac{dx}{d\theta }=3a{\cos }^2\theta (-\sin \theta )$
$\Rightarrow \frac{dy}{d\theta }=3a{\sin }^2\theta (\cos \theta )$
$\Rightarrow \frac{dy}{dx}=\frac{dy}{d\theta }\times \frac{d\theta }{dx}$
$=\left(3a{\sin }^2\theta \cdot \cos \theta \right)\cdot \frac{1}{-\left(3a\sin \theta \cdot {\cos }^2\theta \right)}$
$\frac{dy}{dx}=-\frac{\sin \theta }{\cos \theta },\frac{dy}{dx}=-\tan \theta$
$\Rightarrow \frac{dy}{dx}=-\frac{(y/a{)}^{-1/3}}{(x/a{)}^{1/3}}$
$=-{\left[\frac{y}{a}\times \frac{a}{x}\right]}^{1/3}$
$\frac{dy}{dx}=-{\left(\frac{y}{x}\right)}^{-1/3}=-\sqrt[3]{\frac{y}{x}}$