Question

If \( a \text{ and } b \) are A.M. and G.M. of \( x \text{ and } y \) respectively, then \( x^2 + y^2 \) is equal to: 
 

• A. \( 4a^2 - 2b^2 \)
• B. \( 4a^2 - b^2 \)
• C. \( 2a^2 - 3b^2 \)
• D. \( a^2 - 2b^2 \)
• E. \( 4a^2 - 3b^2 \)

Hint:

Remember the identity \( (x + y)^2 = x^2 + y^2 + 2xy \) and use the given relations for A.M. and G.M. to express the sum of squares in terms of the means.

Answer 1

The Correct Option is A

We are given that \( a \) and \( b \) are the A.M. and G.M. of \( x \) and \( y \), respectively. 
This means:
\[ a = \frac{x + y}{2} \quad {and} \quad b = \sqrt{xy}. \] Step 1: We need to express \( x^2 + y^2 \) in terms of \( a \) and \( b \).
Step 2: We know that: \[ (x + y)^2 = x^2 + y^2 + 2xy. \] Thus, \[ x^2 + y^2 = (x + y)^2 - 2xy. \] Step 3: Substitute the values for \( x + y \) and \( xy \) using \( a \) and \( b \): \[ x^2 + y^2 = (2a)^2 - 2b^2. \] Simplify: \[ x^2 + y^2 = 4a^2 - 2b^2. \] 
Thus, the correct answer is \( 4a^2 - 2b^2 \).
Therefore, the correct answer is option (A).