Question

The roots of \( ax^2 + x + 1 = 0 \), where \( a \neq 0 \), are in the ratio \( 1:1 \). Then \( a \) is equal to:

• A. \( \frac{1}{4} \)
• B. \( \frac{1}{2} \)
• C. \( \frac{3}{4} \)
• D. \( 1 \)
• E. \( 0 \)

Hint:

Equal roots \(\Rightarrow D=0\). Use this directly to save time in exams.

Answer 1

The Correct Option is A

Concept: If the ratio of roots is \( 1:1 \), then the roots are equal. For a quadratic equation, equal roots occur when the discriminant is zero: \[ D = B^2 - 4AC = 0 \]

Step 1: Identify coefficients.
Given equation: \[ ax^2 + x + 1 = 0 \] Comparing with \( Ax^2 + Bx + C = 0 \): \[ A = a, \quad B = 1, \quad C = 1 \]

Step 2: Apply condition for equal roots.
\[ D = B^2 - 4AC = 0 \] \[ 1^2 - 4(a)(1) = 0 \]

Step 3: Solve for \( a \).
\[ 1 - 4a = 0 \] \[ 4a = 1 \] \[ a = \frac{1}{4} \]

Step 4: Final answer.
\[ \boxed{a = \frac{1}{4}} \]