The equivalent resistance of two resistor connected in series is `6 Omega` and their equivalent resistance is `(4)/(3)Omega`. What are the values of resistances ?

Options

A. `4 Omega, 6 Omega`
B. `8 Omega, 1 Omega`
C. `4 Omega, 2 Omega`
D. `6 Omega, 2 Omega`

Answer

To solve the problem of finding the values of two resistors connected in series and parallel, we can follow these steps:

### Step 1: Set up the equations based on the given information
Let the two resistors be \( R_1 \) and \( R_2 \).

From the problem, we know:
1. When connected in series, the equivalent resistance is given by:
\[ R_1 + R_2 = 6 \, \Omega \quad \text{(Equation 1)} \]
2. When connected in parallel, the equivalent resistance is given by:
\[ \frac{R_1 R_2}{R_1 + R_2} = \frac{4}{3} \, \Omega \quad \text{(Equation 2)} \]

### Step 2: Substitute Equation 1 into Equation 2
From Equation 1, we can express \( R_1 + R_2 \) as 6:
\[ \frac{R_1 R_2}{6} = \frac{4}{3} \]
Now, we can cross-multiply to eliminate the fraction:
\[ R_1 R_2 = 6 \cdot \frac{4}{3} \]
Calculating the right side:
\[ R_1 R_2 = 8 \, \Omega \quad \text{(Equation 3)} \]

### Step 3: Solve the system of equations
Now we have two equations:
1. \( R_1 + R_2 = 6 \) (Equation 1)
2. \( R_1 R_2 = 8 \) (Equation 3)

We can express \( R_2 \) in terms of \( R_1 \) from Equation 1:
\[ R_2 = 6 - R_1 \]
Substituting this into Equation 3:
\[ R_1 (6 - R_1) = 8 \]
Expanding this gives:
\[ 6R_1 - R_1^2 = 8 \]
Rearranging it into standard quadratic form:
\[ R_1^2 - 6R_1 + 8 = 0 \]

### Step 4: Factor the quadratic equation
Now we can factor the quadratic equation:
\[ (R_1 - 2)(R_1 - 4) = 0 \]
This gives us the solutions:
\[ R_1 = 2 \, \Omega \quad \text{or} \quad R_1 = 4 \, \Omega \]

### Step 5: Find the corresponding values of \( R_2 \)
Using \( R_1 + R_2 = 6 \):
- If \( R_1 = 2 \, \Omega \):
\[ R_2 = 6 - 2 = 4 \, \Omega \]
- If \( R_1 = 4 \, \Omega \):
\[ R_2 = 6 - 4 = 2 \, \Omega \]

Thus, the values of the resistors are:
\[ R_1 = 2 \, \Omega \quad \text{and} \quad R_2 = 4 \, \Omega \]

### Final Answer
The values of the resistances are \( 2 \, \Omega \) and \( 4 \, \Omega \).

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