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Sum of product of binomial coefficients

If a : b = ...

Question

If $a : b = c : d$ , prove that.
$p a^{2} + q b^{2} : p a - q b^{2} = p c^{2} + q d^{2} : p c^{2} - q d^{2}$ .

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Solution

$\frac{a}{b} = \frac{c}{d} ⟹ \frac{a^{2}}{b^{2}} = \frac{c^{2}}{d^{2}} ⟹ \frac{p a^{2}}{q b^{2}} = \frac{p c^{2}}{q d^{2}}$

Using componendo-dividendo property

$\frac{p a^{2} + q b^{2}}{p a^{2} - q b^{2}} = \frac{p c^{2} + q d^{2}}{p c^{2} - q d^{2}}$

Hence Proved

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