Question

If two positive numbers are in the ratio \( 3 + 2\sqrt{2} : 3 - 2\sqrt{2} \), then the ratio between their A.M. and G.M. is:

• A. \( 6 : 1 \)
• B. \( 3 : 2 \)
• C. \( 2 : 1 \)
• D. \( 3 : 1 \)
• E. \( 1 : 6 \)

Hint:

If the ratio of two numbers is \( \frac{x + \sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}} \), the ratio of their A.M. to G.M. is always \( x : 1 \). Here, \( 3^2 - (2\sqrt{2})^2 = 1 \), so the ratio is \( 3 : 1 \).

Answer 1

The Correct Option is D

Concept: For two positive numbers \( a \) and \( b \):
• Arithmetic Mean (A.M.) = \( \frac{a + b}{2} \)
• Geometric Mean (G.M.) = \( \sqrt{ab} \) Given the ratio \( a : b = (3 + 2\sqrt{2}) : (3 - 2\sqrt{2}) \).

Step 1: Express \( a \) and \( b \) in terms of a constant \( k \).
Let \( a = k(3 + 2\sqrt{2}) \) and \( b = k(3 - 2\sqrt{2}) \).

Step 2: Calculate A.M. and G.M.
\[ \text{A.M.} = \frac{k(3 + 2\sqrt{2} + 3 - 2\sqrt{2})}{2} = \frac{6k}{2} = 3k \] \[ \text{G.M.} = \sqrt{k(3 + 2\sqrt{2}) \cdot k(3 - 2\sqrt{2})} = \sqrt{k^2 (3^2 - (2\sqrt{2})^2)} \] \[ \text{G.M.} = \sqrt{k^2 (9 - 8)} = \sqrt{k^2 \cdot 1} = k \]

Step 3: Find the ratio A.M. : G.M.
\[ \text{Ratio} = \frac{3k}{k} = \frac{3}{1} = 3 : 1 \]