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))/(alphabeta)`
D. `(beta)/(alpha)`

Answer

To solve the problem, we need to analyze the behavior of two resistors \( R_1 \) and \( R_2 \) made of different materials when the temperature changes. The temperature coefficients of the materials are given as \( \alpha \) for \( R_1 \) and \( -\beta \) for \( R_2 \).

### Step-by-Step Solution:

1. **Understand the Change in Resistance with Temperature:**
The resistance of a material changes with temperature according to the formula:
\[ R' = R(1 + \alpha \Delta T) \]
where \( R' \) is the new resistance, \( R \) is the original resistance, \( \alpha \) is the temperature coefficient, and \( \Delta T \) is the change in temperature.

2. **Calculate the New Resistances:**
For \( R_1 \):
\[ R_1' = R_1(1 + \alpha \Delta T) \]
For \( R_2 \):
\[ R_2' = R_2(1 - \beta \Delta T) \]

3. **Determine the Total Resistance in Series:**
The total resistance \( R_{\text{eq}} \) of the series combination of \( R_1 \) and \( R_2 \) is:
\[ R_{\text{eq}} = R_1' + R_2' = R_1(1 + \alpha \Delta T) + R_2(1 - \beta \Delta T) \]

4. **Set Up the Equation for No Change in Resistance:**
For the total resistance to remain constant with temperature, we need:
\[ R_1(1 + \alpha \Delta T) + R_2(1 - \beta \Delta T) = R_1 + R_2 \]

5. **Simplify the Equation:**
Expanding the left side:
\[ R_1 + R_1 \alpha \Delta T + R_2 - R_2 \beta \Delta T = R_1 + R_2 \]
This simplifies to:
\[ R_1 \alpha \Delta T - R_2 \beta \Delta T = 0 \]

6. **Factor Out \( \Delta T \):**
Since \( \Delta T \) cannot be zero (we are considering a change in temperature), we can divide both sides by \( \Delta T \):
\[ R_1 \alpha = R_2 \beta \]

7. **Find the Ratio of Resistances:**
Rearranging gives us:
\[ \frac{R_1}{R_2} = \frac{\beta}{\alpha} \]

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