Four resistances `10 Omega, 5 Omega, 7 Omega` and `3 Omega` are connected so that they form the sides of a rectangle `AB, BC, CD` and `DA` respectively. Another resistance of `10 Omega` is connected across the diagonal `AC`. The equivalent resistance between `A` and `B` is

Options

A. `2 Omega`
B. `5Omega`
C. `7Omega`
D. `10Omega`

Answer

To find the equivalent resistance between points A and B in the given circuit with resistances forming a rectangle and an additional resistance across the diagonal, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Configuration**:
- We have a rectangle with resistances:
- AB = 10 Ω
- BC = 5 Ω
- CD = 7 Ω
- DA = 3 Ω
- A resistance of 10 Ω is connected across the diagonal AC.

2. **Draw the Circuit**:
- Draw a rectangle ABCD.
- Label the resistances on each side:
- AB (10 Ω),
- BC (5 Ω),
- CD (7 Ω),
- DA (3 Ω).
- Draw the diagonal AC and place the 10 Ω resistor across it.

3. **Combine Series Resistors**:
- The resistors CD (7 Ω) and DA (3 Ω) are in series.
- Calculate the equivalent resistance (R1) of these two:
\[ R_1 = R_{CD} + R_{DA} = 7 \, \Omega + 3 \, \Omega = 10 \, \Omega \]

4. **Redraw the Circuit**:
- Replace the series combination of 7 Ω and 3 Ω with a single 10 Ω resistor.
- Now the circuit has:
- AB = 10 Ω,
- BC = 5 Ω,
- R1 (CD + DA) = 10 Ω,
- AC = 10 Ω.

5. **Identify Parallel Resistors**:
- The resistors across the diagonal AC (10 Ω) and the equivalent resistor R1 (10 Ω) are in parallel.
- Calculate the equivalent resistance (R2) of these two:
\[ \frac{1}{R_2} = \frac{1}{R_{AC}} + \frac{1}{R_1} = \frac{1}{10 \, \Omega} + \frac{1}{10 \, \Omega} = \frac{2}{10} = \frac{1}{5} \]
\[ R_2 = 5 \, \Omega \]

6. **Redraw the Circuit Again**:
- Now we have:
- AB = 10 Ω,
- BC = 5 Ω,
- R2 (parallel combination) = 5 Ω.

7. **Combine Series Resistors Again**:
- The resistors R2 (5 Ω) and the resistor AB (10 Ω) are in series.
- Calculate the total equivalent resistance (R_eq):
\[ R_{eq} = R_{AB} + R_2 = 10 \, \Omega + 5 \, \Omega = 15 \, \Omega \]

### Final Answer:
The equivalent resistance between points A and B is **15 Ω**.