Question

If (4a + 5b) (4c – 5d) = (4a – 5d) (4c + 5d), prove that a, b, c, d are in proporton.

Answer 1

(4a + 5b) (4c – 5d) = (4a – 5d) (4c + 5d)

⇒ `(4a + 5b)/(4a - 5b) = (4c + 5d)/(4c - 5d)`

Applying componendo and dividendo
`(4a + 5b + 4a - 5b)/(4a + 5b - 4a + 5b) = (4c + 5d + 4c - 5d)/(4c + 5d - 4c + 5d)`

⇒ `(8a)/(10b) = (8c)/(10d)`

⇒ `a/b = c/d`

Hence, a, b, c, d are in proportion.