Question

If x and y be unequal and x: y is the duplicate ratio of (x + z) and (y + z) prove that z is mean proportional between x and y.

Answer 1

Since x : y is duplicate ratio of (x + z) and (y + z)
∴ x : y = (x + z)² : (y + z)²
x(y + z)² = y(x + z)²
On simplifying we get
xy² + xz² = yx² + yz²
⇒ x²y - xy² = xz² - yz²
⇒ xy(x -y) = z²(x - y)
⇒ xy = z²
⇒ x : z : : z : y.
Hence proved.