If one roots of the equation `(x^(2))/(a) + (x)/(b) + (1)/(c ) = 0` is reciprocal of the other, then which one of the following is correct?

Options

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

Answer

To solve the problem, we need to analyze the given quadratic equation and the conditions provided. The equation is:

\[ \frac{x^2}{a} + \frac{x}{b} + \frac{1}{c} = 0 \]

We are told that one root of this equation is the reciprocal of the other. Let's denote the roots of the equation as \(\alpha\) and \(\beta\). Given that \(\beta = \frac{1}{\alpha}\), we can use the properties of the roots of a quadratic equation.

### Step 1: Identify the sum and product of the roots
For a quadratic equation of the form \(Ax^2 + Bx + C = 0\), the sum of the roots (\(\alpha + \beta\)) and the product of the roots (\(\alpha \beta\)) can be expressed as:

\[ \alpha + \beta = -\frac{B}{A} \]
\[ \alpha \beta = \frac{C}{A} \]

In our case:
- \(A = \frac{1}{a}\)
- \(B = \frac{1}{b}\)
- \(C = \frac{1}{c}\)

Thus, we can find:

\[ \alpha + \beta = -\frac{\frac{1}{b}}{\frac{1}{a}} = -\frac{a}{b} \]

\[ \alpha \beta = \frac{\frac{1}{c}}{\frac{1}{a}} = \frac{a}{c} \]

### Step 2: Set up the equation for reciprocal roots
Since \(\beta = \frac{1}{\alpha}\), we know that:

\[ \alpha \beta = 1 \]

From the product of the roots we derived earlier, we have:

\[ \frac{a}{c} = 1 \]

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

\[ a = c \]

### Conclusion
Thus, the correct relationship among the variables is:

\[ a = c \]

### Final Answer
The correct option is that \(a\) is equal to \(c\).

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