Two conducting rods when connected between two points at constant but different temperatures separately, the rate of heat flow through them is `q_1 and q_2`.

Options

A. When they are connected in series, the net rate of heat flow is `q_1 + q_2`
B. When they are connected in series, the net rate of heat flow is `((q_1q_2)/(q_1+q_2))`
C. When they are connected in parallel, the net rate of heat flow is `q_1 + q_2`
D. When they are connected in parallel, the net rate of heat flow is `((q_1q_2)/(q_1+q_2))`

Explanation

To solve the problem regarding the rate of heat flow through two conducting rods connected at constant but different temperatures, we will analyze the situation in both series and parallel configurations.

### Step-by-Step Solution:

1. **Identify the Given Variables:**
- Let \( q_1 \) be the rate of heat flow through the first rod.
- Let \( q_2 \) be the rate of heat flow through the second rod.
- Let \( R_1 \) be the resistance of the first rod.
- Let \( R_2 \) be the resistance of the second rod.
- Let \( \theta \) be the temperature difference across the rods.

2. **Use the Formula for Heat Flow:**
The rate of heat flow through a conductor is given by the formula:
\[ q = \frac{\theta}{R} \]
Therefore, for the first rod:
\[ q_1 = \frac{\theta}{R_1} \]
And for the second rod:
\[ q_2 = \frac{\theta}{R_2} \]

3. **Case 1: Rods Connected in Series:**
- When the rods are connected in series, the effective resistance \( R_{\text{eff}} \) is:
\[ R_{\text{eff}} = R_1 + R_2 \]
- The total heat flow \( q_{\text{series}} \) through the series connection is:
\[ q_{\text{series}} = \frac{\theta}{R_{\text{eff}}} = \frac{\theta}{R_1 + R_2} \]
- Using the expressions for \( q_1 \) and \( q_2 \), we can express this as:
\[ q_{\text{series}} = \frac{q_1 q_2}{q_1 + q_2} \]

4. **Case 2: Rods Connected in Parallel:**
- When the rods are connected in parallel, the effective resistance \( R_{\text{eff}} \) is:
\[ \frac{1}{R_{\text{eff}}} = \frac{1}{R_1} + \frac{1}{R_2} \]
- This can be rewritten as:
\[ R_{\text{eff}} = \frac{R_1 R_2}{R_1 + R_2} \]
- The total heat flow \( q_{\text{parallel}} \) through the parallel connection is:
\[ q_{\text{parallel}} = \frac{\theta}{R_{\text{eff}}} = \frac{\theta (R_1 + R_2)}{R_1 R_2} \]
- Again, using the expressions for \( q_1 \) and \( q_2 \), we can express this as:
\[ q_{\text{parallel}} = q_1 + q_2 \]

5. **Conclusion:**
- For the series connection, the heat flow is given by:
\[ q_{\text{series}} = \frac{q_1 q_2}{q_1 + q_2} \]
- For the parallel connection, the heat flow is given by:
\[ q_{\text{parallel}} = q_1 + q_2 \]

### Final Answer:
- The correct expressions for heat flow are:
- In series: \( q_{\text{series}} = \frac{q_1 q_2}{q_1 + q_2} \)
- In parallel: \( q_{\text{parallel}} = q_1 + q_2 \)