If x/y = 1/3, then (x² + y²)/(x² - y²) = ?

Question: If x/y = 1/3, then (x² + y²)/(x² - y²) = ?

Solution:
(x² + y²)/(x² - y²)
= {(x² + y²)/y²}/{x² - y²)/y² [Dividing the numerator and denominator by y²]
= {(x²/y²) + 1}/{(x²/y²) - 1}
= {(x/y)² + 1}/{(x/y)² - 1)}
= {(1/3)² + 1}/{(1/3)² - 1} [given, x/y = 1/3]
= {(1/9) + 1}/{(1/9} - 1}
= (10/9)/(-8/9)
= (10/9) × (9/-8)
= - 5/4

Shortcut:
Take x = 1, y = 3 (because 1/3 = 1/3)
∴ (x² + y²)/(x² - y²) = (1² + 3² )/(1² - 3²)
= (1 + 9)/(1 - 9) = - 5/4