Question

If a: b with a ≠ b is the duplicate ratio of a + c: b + c, show that c² = ab.

Answer 1

The duplicate ratio of
`a/b = ((a + c)^2)/((b + c)^2)`
⇒ `a/b = (a^2 + c^2 + 2ac)/(b^2 + c^2 + 2bc)`
⇒ ab² + ac² + 2abc = a²b + bc² + 2abc
⇒ ac² - bc² = a²b - ab²
⇒ c²(a - b) = ab(a - b)
Hence, c² = ab.
Hence proved.