Eliminate θ between the given equations:

x = a sec³ θ, y = b tan³ θ

Given,

⇒ x = a sec³ θ

⇒ sec³ θ = $\dfrac{x}{a}$

⇒ sec² θ = $\Big(\dfrac{x}{a}\Big)^\dfrac{2}{3}$

⇒ y = b tan³ θ

⇒ tan³ θ = $\dfrac{y}{b}$

⇒ tan² θ = $\Big(\dfrac{y}{b}\Big)^\dfrac{2}{3}$

Using the identity

sec² θ - tan² θ = 1

Substitute,

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

Hence,the required relation is $\Big(\dfrac{x}{a}\Big)^\dfrac{2}{3} - \Big(\dfrac{y}{b}\Big)^\dfrac{2}{3} = 1$.