Question

If the A.M. and G.M. of the roots of a quadratic equation in \( x \) are \( p \) and \( q \) respectively, then its equation is

• A. \( x^2 + 2px + q^2 = 0 \)
• B. \( x^2 + px + q^2 = 0 \)
• C. \( x^2 - px + q^2 = 0 \)
• D. \( x^2 - 2px + q^2 = 0 \)

Hint:

For a quadratic equation with given A.M. and G.M., use the relationships between the roots and coefficients to construct the equation.

Answer 1

The Correct Option is D

Step 1: Use the relationships between the roots and coefficients.
Let the roots of the quadratic equation be \( \alpha \) and \( \beta \). The A.M. (Arithmetic Mean) and G.M. (Geometric Mean) are given by: \[ A.M. = \frac{\alpha + \beta}{2} = p, \quad G.M. = \sqrt{\alpha \beta} = q \]
Step 2: Set up the equation using these relationships.
From the A.M., we have \( \alpha + \beta = 2p \), and from the G.M., we have \( \alpha \beta = q^2 \). The equation of the quadratic is: \[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \] Substitute \( \alpha + \beta = 2p \) and \( \alpha \beta = q^2 \): \[ x^2 - 2px + q^2 = 0 \]
Step 3: Conclusion.
The equation is \( x^2 - 2px + q^2 = 0 \).