If $(a - b) : (a + b) = 1 : 5$ , then what is $(a^{2} - b^{2}) : (a^{2} + b^{2})$ equal to?

Question

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Answer 1

Given, $\frac{a - b}{a + b} = \frac{1}{5}$
$\Rightarrow 5 a - 5 b = a + b$
$\Rightarrow 4 a = 6 b$
$\Rightarrow \frac{a}{b} = \frac{3}{2}$ i.e. $\frac{a}{3} = \frac{b}{2} = k$
Therefore, $\frac{a ^{2} - b ^{2}}{a ^{2} + b ^{2}} = \frac{9 k ^{2} - 4 k ^{2}}{9 k ^{2} + 4 k ^{2}} = \frac{5}{1 3}$
Ths the required ratio is $5 : 13$ .

Answer 2

Given,

\frac{a - b}{a + b} = \frac{1}{5}

\Rightarrow 5 a - 5 b = a + b

\Rightarrow 4 a = 6 b

\Rightarrow \frac{a}{b} = \frac{3}{2}

i.e.

\frac{a}{3} = \frac{b}{2} = k

Therefore,

\frac{a ^{2} - b ^{2}}{a ^{2} + b ^{2}} = \frac{9 k ^{2} - 4 k ^{2}}{9 k ^{2} + 4 k ^{2}} = \frac{5}{1 3}

Ths the required ratio is

5 : 13

.