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