If one root of the quadratic equation `ax^2+bx+c=0` is the reciprocal of the other, then which of the following is correct?

Options

A. a=c
B. ac=1
C. b=c
D. a=b

Answer

To solve the problem, we need to analyze the properties of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) when one root is the reciprocal of the other.

### Step-by-Step Solution:

1. **Define the Roots**:
Let the two roots of the quadratic equation be \( \alpha \) and \( \frac{1}{\alpha} \).

2. **Sum of the Roots**:
According to Vieta's formulas, the sum of the roots of the quadratic equation is given by:
\[ \alpha + \frac{1}{\alpha} = -\frac{b}{a} \]

3. **Product of the Roots**:
The product of the roots is also given by Vieta's formulas:
\[ \alpha \cdot \frac{1}{\alpha} = 1 = \frac{c}{a} \]

4. **Relate \( c \) and \( a \)**:
From the product of the roots, we have:
\[ \frac{c}{a} = 1 \implies c = a \]

5. **Conclusion**:
We have established two relationships:
- \( c = a \)
- \( \alpha + \frac{1}{\alpha} = -\frac{b}{a} \)

However, the key relationship derived from the condition that one root is the reciprocal of the other is \( c = a \).

### Final Answer:
Thus, the correct statement is that \( c = a \).

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