If x = a cos³ θ and y = b sin³ θ, then $\Big(\dfrac{x}{a}\Big)^{\dfrac{2}{3}} + \Big(\dfrac{y}{b}\Big)^{\dfrac{2}{3}}$ is equal to :

1. a

2. b

3. 1

4. 2

Given,

x = a cos³ θ and y = b sin³ θ

$\Rightarrow \dfrac{x}{a} = \cos^3 θ \\[1em] \Rightarrow \Big(\dfrac{x}{a}\Big)^{\dfrac{2}{3}} = \cos^2 θ \\[1em] \Rightarrow \dfrac{y}{b} = \sin^3 θ \\[1em] \Rightarrow \Big(\dfrac{y}{b}\Big)^{\dfrac{2}{3}} = \sin^2 θ$

Add the expressions,

$\Big(\dfrac{x}{a}\Big)^{\dfrac{2}{3}} + \Big(\dfrac{y}{b}\Big)^{\dfrac{2}{3}}$ = cos² θ + sin² θ

$\Big(\dfrac{x}{a}\Big)^{\dfrac{2}{3}} + \Big(\dfrac{y}{b}\Big)^{\dfrac{2}{3}}$ = 1

Hence, option 3 is the correct option.