If (a + b): (a-b) = 5 : 3, then find ( \( \Large a^{2}+b^{2} \)) : ( \( \Large a^{2}-b^{2} \)).

\( \Large \frac{a+b}{a-b} = \frac{5}{3} \) => 3a + 3b = 5a - 5b

=> 2a = 8b => a = 4-b

=> \( \Large \frac{a}{b} = \frac{4}{1} \)

Now \( \Large \frac{ \left(a^{2} + b^{2}\right) }{ \left(a^{2} - b^{2}\right) } = \frac{\frac{a^{2}}{b^{2}} + 1 }{ \frac{a^{2}}{b^{2}} - 1 } \)

\( \Large = \frac{ \left(\frac{a}{b}\right)^{2} + 1 }{ \left(\frac{a}{b}\right)^{2} - 1 } \)

\( \Large = \frac{ \left(\frac{4}{1}\right)^{2} + 1 }{ \left(\frac{4}{1}\right)^{2} - 1 } \)

=> \( \Large \frac{16 + 1}{ 16 - 1} = \frac{17}{15} \)

\( \Large \left(a^{2} + b^{2}\right) : \left(a^{2} - b^{2}\right) = 17 : 15 \)