Question

If x = a cos³θ and y = b sin³θ, prove that ${\left(\frac{x}{a}\right)}^{2/3}+{\left(\frac{y}{b}\right)}^{2/3}=1.$

Answer 1

$\begin{array}{l}\\ \text{We have}x=a{\cos }^3\theta \\ \text{=>}\frac{x}{a}={\cos }^3\theta ...\left(\text{i}\right)\\ \text{Again,}y=b{\sin }^3\theta \\ \text{=>}\frac{y}{b}={\sin }^3\theta ...\left(\text{ii}\right)\\ \text{Now, LHS=}{\left(\frac{x}{a}\right)}^{\frac{2}{3}}+{\left(\frac{y}{b}\right)}^{\frac{2}{3}}\\ \\ =({\cos }^3\theta {)}^{\frac{2}{3}}+{\left({\sin }^3\theta \right)}^{\frac{2}{3}}\text{[From}\left(\text{i}\right)\text{and}\left(\text{ii}\right)]\\ {\text{=cos}}^2\theta +{\sin }^2\theta \\ =1\\ \text{Hence, LHS}=\text{RHS}\end{array}$