If (4a² + 7b²) : (4a² − 7b²) = (4c² + 7d²) : (4c² − 7d²), prove that a : b = c : d.

Given,

(4a² + 7b²) : (4a² − 7b²) = (4c² + 7d²) : (4c² − 7d²)

$\Rightarrow \dfrac{4a^2 + 7b^2}{4a^2 - 7b^2} = \dfrac{4c^2 + 7d^2}{4c^2 - 7d^2}$

Applying componendo and dividendo:

$\Rightarrow \dfrac{4a^2 + 7b^2 + 4a^2 - 7b^2}{4a^2 + 7b^2 - (4a^2 - 7b^2)} = \dfrac{4c^2 + 7d^2 + 4c^2 - 7d^2}{4c^2 + 7d^2 - (4c^2 - 7d^2)} \\[1em] \Rightarrow \dfrac{8a^2}{4a^2 + 7b^2 - 4a^2 + 7b^2} = \dfrac{8c^2}{4c^2 + 7d^2 - 4c^2 + 7d^2} \\[1em] \Rightarrow \dfrac{8a^2}{14b^2} = \dfrac{8c^2}{14d^2} \\[1em] \Rightarrow \dfrac{a^2}{b^2} = \dfrac{c^2}{d^2} \\[1em] \Rightarrow \sqrt{\dfrac{a^2}{b^2}} = \sqrt{\dfrac{c^2}{d^2}} \\[1em] \Rightarrow \dfrac{a}{b} = \dfrac{c}{d}.$

Hence, proved that a : b = c : d.