Question

If the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are in the ratio 3:4, then which of the following relationships holds between the coefficients?

• A. \( 7b^2 = 48ac \)
• B. \( 48b^2 = 7ac \)
• C. \( b^2 = 12ac \)
• D. \( 12b^2 = ac \)

Hint:

\textbf{Key Fact:} For a quadratic equation with roots in a given ratio, use the sum and product of roots to relate the coefficients.

Answer 1

The Correct Option is A

• Define the Roots: Let the roots of the quadratic equation \( ax^2 + bx + c = 0 \) be \( \alpha \) and \( \beta \), with \( \alpha : \beta = 3 : 4 \). Thus, \( \alpha = 3k \) and \( \beta = 4k \), where \( k \) is a constant.
• Use Sum and Product of Roots:
  • Sum of roots: \( \alpha + \beta = 3k + 4k = 7k = -\frac{b}{a} \), so \( 7k = -\frac{b}{a} \), or \( k = -\frac{b}{7a} \).
  • Product of roots: \( \alpha \beta = (3k)(4k) = 12k^2 = \frac{c}{a} \), so \( 12k^2 = \frac{c}{a} \).
• Substitute \( k \): Substitute \( k = -\frac{b}{7a} \) into the product equation: \[ 12 \left( -\frac{b}{7a} \right)^2 = \frac{c}{a}. \]
• Simplify: \[ 12 \cdot \frac{b^2}{49a^2} = \frac{c}{a}. \] Multiply both sides by \( 49a^2 \): \[ 12 b^2 = 49 a c. \] Rearranged, \[ 7 b^2 = 48 a c. \]
• Conclusion: The correct answer is (1) \( 7 b^2 = 48 a c \).