If the A.M. of two positive numbers a and b (a b) is twice their geometric mean. Prove that : a : b = (2 + ) : (2 – ).

Question

If the A.M. of two positive numbers a and b (a > b) is twice their geometric mean. Prove that : a : b = (2 + ) : (2 – ).

Philoid · Accepted Answer

Let the two numbers be a and b.

GM = √ab

According to the given condition,

⇒

a + b = 4√ab …(1)

(a + b)² = 16ab

Also,

(a – b)² = (a + b)² – 4ab

= 16ab – 4ab

= 12ab

⇒ a – b = 2√3ab…(2)

Adding (1) and (2), we obtain

2a = (4 + 2√3 )√ab

a = (2 + √3)√ab

substituting the value of a in (1), we obtain,

b =(2 – √3)√ab

∴

Thus, the required ratio is (2+√3) : (2–√3).