Question

If r² = pq, show that p : q is the duplicate ratio of (p + r) : (q + r).

Answer 1

Given, r² = pq

Duplicate ratio of (p + r) : (q + r)

= (p + r)² : (q + r)²

= (p² + r² + 2pr) : (q² + r² + 2qr)

= (p² + pq + 2pr) : (q² + pq + 2qr)

= p(p + q + 2r) : q(q + p + 2r)

= p : q

Thus, p : q is the duplicate ratio of (p + r) :  (q + r).