Two resistances `R_1` and `R_2` are made of different materials. The temperature coefficient of the material of `R_1` is `alpha` and of the material of `R_2` is `-beta`. The resistance of the series combination of `R_1` and `R_2` will not change with temperature, if `R_1//R_2` equals.

Options

A. `(alpha)/(beta)`
B. `(alpha+beta)/(alpha - beta)`
C. `(alpha^2 + beta^2)/(alpha beta)`
D. `(beta)/(alpha)`

Explanation

To solve the problem, we need to determine the condition under which the total resistance of two resistors in series, \( R_1 \) and \( R_2 \), remains constant with temperature changes. The resistors have different temperature coefficients of resistance, \( \alpha \) for \( R_1 \) and \( -\beta \) for \( R_2 \).

### Step-by-Step Solution:

1. **Understand the Change in Resistance**:
The change in resistance due to temperature for \( R_1 \) can be expressed as:
\[ \Delta R_1 = R_1 \cdot \alpha \cdot \Delta T \]
For \( R_2 \), since its temperature coefficient is negative, the change in resistance is:
\[ \Delta R_2 = R_2 \cdot (-\beta) \cdot \Delta T = -R_2 \cdot \beta \cdot \Delta T \]

2. **Total Resistance in Series**:
The total resistance \( R \) when \( R_1 \) and \( R_2 \) are connected in series is:
\[ R = R_1 + R_2 \]
For the total resistance to remain constant with temperature, the sum of the changes in resistance must equal zero:
\[ \Delta R_1 + \Delta R_2 = 0 \]

3. **Set Up the Equation**:
Substitute the expressions for \( \Delta R_1 \) and \( \Delta R_2 \) into the equation:
\[ R_1 \cdot \alpha \cdot \Delta T - R_2 \cdot \beta \cdot \Delta T = 0 \]
Since \( \Delta T \) is common in both terms, we can cancel it out (assuming \( \Delta T \neq 0 \)):
\[ R_1 \cdot \alpha - R_2 \cdot \beta = 0 \]

4. **Rearranging the Equation**:
Rearranging the equation gives:
\[ R_1 \cdot \alpha = R_2 \cdot \beta \]

5. **Finding the Ratio**:
To find the ratio \( \frac{R_1}{R_2} \), we can rearrange the equation:
\[ \frac{R_1}{R_2} = \frac{\beta}{\alpha} \]

### Final Answer:
Thus, the condition under which the resistance of the series combination of \( R_1 \) and \( R_2 \) does not change with temperature is:
\[ \frac{R_1}{R_2} = \frac{\beta}{\alpha} \]