If the A.M. of two positive numbers a and b (a > b) is twice their geo

1.	If the A.M. of two positive numbers a and b (a > b) is twice their geometric mean. Prove that : a : b = (2 + √3) : (2 – √3).
Answer» Let the two numbers be a and b. GM = √ab According to the given condition, ⇒ a+b/2 = 2√ab ⇒ 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 \(\therefore\frac{a}{b}=\frac{2+\sqrt3}{2-\sqrt3}\) ∴ Thus, the required ratio is (2+√3) : (2–√3).

If the A.M. of two positive numbers a and b (a > b) is twice their geometric mean. Prove that : a : b = (2 + √3) : (2 – √3).

Answer»

Let the two numbers be a and b.

GM = √ab

According to the given condition,

⇒ a+b/2 = 2√ab

⇒ 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

\(\therefore\frac{a}{b}=\frac{2+\sqrt3}{2-\sqrt3}\)

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