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

Answer

To solve the problem step by step, we start with the given quadratic equation and the condition about its roots.

### Step 1: Understand 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. We can express this relationship as:
\[ \alpha = \frac{1}{\beta} \]

**Hint:** Recall that if one root is the reciprocal of the other, their product is equal to 1.

### Step 2: Use the product of roots
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the product of the roots \( \alpha \) and \( \beta \) can be given by the formula:
\[ \alpha \cdot \beta = \frac{c}{a} \]
Since we established that \( \alpha \cdot \beta = 1 \), we can set up the equation:
\[ 1 = \frac{c}{a} \]

**Hint:** This relationship allows us to express \( c \) in terms of \( a \).

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

**Hint:** This shows that \( c \) must equal \( a \) when one root is the reciprocal of the other.

### Step 4: Conclusion
Based on our findings, the correct relationship among the coefficients of the quadratic equation is:
\[ a = c \]
Thus, the correct option is that \( a \) is equal to \( c \).

**Final Answer:** The correct option is \( a = c \).