Question

If (11a² + 13b²) (11c² – 13d²) = (11a² – 13b²)(11c² + 13d²), prove that a : b :: c : d.

Answer 1

(11a² + 13b²) (11c² – 13d²) = (11a² – 13b²)(11c² + 13d²)

⇒ `(11a + 13b^2)/(11a^2 - 13b^2) = (11c^2 + 13d^2)/(11c^2 - 13d^2)`

Applying componendo and dividendo

`(11a^2 + 13b^2 + 11a^2 - 13b^2)/(11a^2 + 13b^2 - 11a^2 + 13b^2) = (11c^2 + 13d^2 + 11c^2 - 13d^2)/(11c^2 + 13d^2 - 11c^2 + 13d^2)`

⇒ `(22a^2)/(26b^2) = (22c^2)/(26d^2)`

⇒ `a^2/b^2 = c^2/d^2    ...("Dividing by" 22/26)`

⇒ `a/b = c/d`
Hence a : b :: c : d.