Question

Eliminate $\theta$, $x=a{\cos }^3\theta y=b{\sin }^3\theta$

Answer 1

$x=a{\cos }^3\theta \Rightarrow {\cos }^3\theta =\frac{x}{a}$ or $\cos \theta ={\left(\frac{x}{a}\right)}^{\frac{1}{3}}$
Similarly $\sin \theta ={\left(\frac{y}{b}\right)}^{\frac{1}{3}}$
$\therefore {\sin }^2\theta +{\cos }^2\theta ={\left(\frac{x}{a}\right)}^{\frac{2}{3}}+{\left(\frac{y}{b}\right)}^{\frac{2}{3}}$
$\because {\left(\frac{x}{a}\right)}^{\frac{2}{3}}+{\left(\frac{y}{b}\right)}^{\frac{2}{3}}=1$