If a, b, c and d are in proportion, the value of $\dfrac{8a^2 - 5b^2}{8c^2 - 5d^2}$ is equal to :

1. a² : b²

2. a² : c²

3. a² : d²

4. c² : d²

Given,

a, b, c and d are in proportion.

$\therefore \dfrac{a}{b} = \dfrac{c}{d}$ = k (let)

∴ a = bk and c = dk

Solving,

$\Rightarrow \dfrac{8a^2 - 5b^2}{8c^2 - 5d^2} \\[1em] \Rightarrow \dfrac{8(bk)^2 - 5b^2}{8(dk)^2 - 5d^2} \\[1em] \Rightarrow \dfrac{8b^2k^2 - 5b^2}{8d^2k^2 - 5d^2} \\[1em] \Rightarrow \dfrac{b^2(8k^2 - 5)}{d^2(8k^2 - 5)} \\[1em] \Rightarrow \dfrac{b^2}{d^2} \space …..(1)$

As,

$\Rightarrow \dfrac{a}{b} = \dfrac{c}{d} \\[1em] \Rightarrow \dfrac{b}{d} = \dfrac{a}{c}$

Substituting value of $\dfrac{b}{d}$ in (1), we get :

$\Rightarrow \dfrac{b^2}{d^2} = \dfrac{a^2}{c^2}$ = a² : c².

Hence, Option 2 is the correct option.