Question

The AM between two positive numbers a and b(a>b) is twice their GM. Prove that a:b = (2+ √3) : (2 - √3)

Answer 1

To prove: Prove that a:b = (2+ √3) : (2 - √3)

Given: Arithmetic mean is twice of Geometric mean. 

Formula used: (i) Arithmetic mean between a and b = \(\frac{a+ b}{2}\)

(ii) Geometric mean between a and b = \(\sqrt{ab}\)

Am = 2(Gm)

\(\frac{a+ b}{2}\) = 2 (\(\sqrt{ab}\))

⇒ a + b = 4\(\sqrt{ab}\) 

Squaring both side 

⇒ (a + b)2 = 16ab … (i) 

We know that (a – b)² = (a + b)² – 4ab 

From eqn. (i) 

⇒ (a – b)² = 16ab – 4ab 

⇒ (a – b)² = 12ab … (ii) 

Dividing eqn. (i) and (ii)

\(\frac{(a + b)^2}{(a-b)^2} = \frac{16ab}{12ab}\)

\(\left(\frac{a+b}{a-b}\right)^2\) = \(\frac{16}{12}\)

Taking square root both side

Applying componendo and dividend

Hence Proved