Question

By eliminating $\theta$, from $x=a{\cos }^3\theta ,y=a{\sin }^3\theta$,the expression is

• A. $\sqrt{x}+\sqrt{y}=\sqrt{a}$
• B. $x^2+y^2=a^2$
• C. $x^{2/3}+y^{2/3}=a^{2/3}$
• D. none of the above

Answer 1

The correct option is C $x^{2/3}+y^{2/3}=a^{2/3}$

$x=a{\cos }^3\theta$
$\therefore (\frac{x}{a}{)}^{\frac{1}{3}}=\cos \theta$
Similarly $(\frac{y}{a}{)}^{\frac{1}{3}}=\sin \theta$
We know, ${\sin }^2\theta +{\cos }^2\theta =1$

$\therefore (\frac{y}{a}{)}^{\frac{2}{3}}+(\frac{x}{a}{)}^{\frac{2}{3}}=1$
Hence,
$y^{\frac{2}{3}}+x^{\frac{2}{3}}=a^{\frac{2}{3}}$
Hence, option 'C' is correct.