Question

If \(α, ẞ\) are the roots of a quadratic equation \( ax^2 + bx+c=0, a ≠0\) then \(\frac{1}{\alpha} + \frac{1}{\beta}\)=___

• A. \(\frac{-b}{a}\)
• B. \(\frac{c}{a}\)
• C. \(\frac{-b}{c}\)
• D. \(\frac{b}{c}\)

Answer 1

The Correct Option is C

To solve this problem, we are given that \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \), and we are asked to find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \).

1. Using Root Relationships:
For the quadratic equation \( ax^2 + bx + c = 0 \), the sum and product of the roots are given by:

• \( \alpha + \beta = -\frac{b}{a} \)
• \( \alpha \beta = \frac{c}{a} \)

2. Using the Identity:
We use the identity:

\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} \]

3. Substituting the Values:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{-\frac{b}{a}}{\frac{c}{a}} = \frac{-b}{c} \]

4. Evaluating the Options:

• (1) \( -\frac{b}{a} \) – Incorrect
• (2) \( \frac{c}{a} \) – Incorrect
• (3) \( -\frac{b}{c} \) – Correct
• (4) \( \frac{b}{c} \) – Incorrect

Final Answer:
The correct answer is (C) \( -\frac{b}{c} \).