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

Options

A. `a=c`
B. `b=c`
C. `a=-c`
D. `b=0`

Explanation

To solve the problem step by step, we need to analyze the condition given in the question.

### Step 1: Understanding the Roots
Let the roots of the quadratic equation \( ax^2 + bx + c = 0 \) be \( \alpha \) and \( \beta \). According to the problem, one root is the reciprocal of the other. Therefore, we can write:
\[ \beta = \frac{1}{\alpha} \]

### Step 2: Using the Product of Roots
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the product of the roots is given by:
\[ \alpha \cdot \beta = \frac{c}{a} \]
Substituting \( \beta \) with \( \frac{1}{\alpha} \):
\[ \alpha \cdot \frac{1}{\alpha} = \frac{c}{a} \]
This simplifies to:
\[ 1 = \frac{c}{a} \]

### Step 3: Rearranging the Equation
From the equation \( 1 = \frac{c}{a} \), we can rearrange it to find a relationship between \( a \) and \( c \):
\[ c = a \]

### Step 4: Conclusion
Thus, we have established that if one root of the equation is the reciprocal of the other, then:
\[ a = c \]
Now, we can check the options provided in the question. The correct option is:
\[ \text{A is equal to C} \]

### Final Answer
The correct condition is \( a = c \).

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