Question

If a : b : : c : d, prove that (2a + 3b)(2c – 3d) = (2a – 3b)(2c + 3d)

Answer 1

∵ a : b :: c : d

∴ `a/b = c/d`

⇒ `(2a)/(3b) = (2c)/(3d)    ...("Multiply by" 2/3)`
Applying componendo and dividendo,
`(2a + 3b)/(2a - 3b) = (2c + 3d)/(2c - 3d)`

⇒ (2a + 3b)(2c – 3d)
= (2a – 3b)(2c + 3d) ...(By corss multiplication).