Two resistance `R_(1)` and `R_(2)` are made of different material. The temperature coefficient of the material of `R_(1)` is `alpha` and of the material of `R_(2)` is `-beta`. Then resistance of the series combination of `R_(1)` and `R_(2)` will not change with temperature, if `R_(1)//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 analyze the conditions under which the resistance of the series combination of two resistors \( R_1 \) and \( R_2 \) remains constant with temperature changes.

### Step-by-Step Solution:

1. **Understanding the Temperature Coefficients**:
- The temperature coefficient of resistance for \( R_1 \) is \( \alpha \) (positive), meaning its resistance increases with temperature.
- The temperature coefficient for \( R_2 \) is \( -\beta \) (negative), meaning its resistance decreases with temperature.

2. **Resistance Change with Temperature**:
- The resistance of \( R_1 \) at temperature \( t \) can be expressed as:
\[ R_1' = R_1 (1 + \alpha t) \]
- The resistance of \( R_2 \) at temperature \( t \) can be expressed as:
\[ R_2' = R_2 (1 - \beta t) \]

3. **Total Resistance in Series**:
- The total resistance \( R_s \) of the series combination of \( R_1 \) and \( R_2 \) is:
\[ R_s = R_1' + R_2' = R_1(1 + \alpha t) + R_2(1 - \beta t) \]

4. **Setting Up the Equation**:
- Substitute the expressions for \( R_1' \) and \( R_2' \):
\[ R_s = R_1(1 + \alpha t) + R_2(1 - \beta t) \]
- Expanding this gives:
\[ R_s = R_1 + R_2 + R_1 \alpha t - R_2 \beta t \]

5. **Condition for Constant Resistance**:
- For the total resistance \( R_s \) to remain constant with temperature, the coefficient of \( t \) must equal zero:
\[ R_1 \alpha - R_2 \beta = 0 \]

6. **Finding the Ratio**:
- Rearranging the equation gives:
\[ R_1 \alpha = R_2 \beta \]
- Dividing both sides by \( R_2 \) and \( \alpha \):
\[ \frac{R_1}{R_2} = \frac{\beta}{\alpha} \]

### Final Result:
Thus, the resistance of the series combination of \( R_1 \) and \( R_2 \) will not change with temperature if:
\[ \frac{R_1}{R_2} = \frac{\beta}{\alpha} \]