The resultant resistance of two resistance in series is `50 Omega `and it is `12 Omega `, when they are in parallel. The individual resistances are

Options

A. `20 Omega and 15 Omega `
B. `15 Omega and 30 Omega `
C. ` 20 Omega and 30 Omega `
D. ` 10 Omega and 15 Omega`

Answer

To find the individual resistances \( R_1 \) and \( R_2 \) given that their resultant resistance in series is \( 50 \, \Omega \) and in parallel is \( 12 \, \Omega \), we can follow these steps:

### Step 1: Set Up the Equations
When resistances are in series, the total resistance \( R_s \) is given by:
\[ R_s = R_1 + R_2 \]
From the problem, we know:
\[ R_1 + R_2 = 50 \quad \text{(Equation 1)} \]

When resistances are in parallel, the total resistance \( R_p \) is given by:
\[ \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} \]
This can be rearranged to:
\[ R_1 R_2 = R_p (R_1 + R_2) \]
Substituting the known values:
\[ R_1 R_2 = 12 \times 50 = 600 \quad \text{(Equation 2)} \]

### Step 2: Solve for One Resistance
From Equation 1, we can express \( R_2 \) in terms of \( R_1 \):
\[ R_2 = 50 - R_1 \]

### Step 3: Substitute into Equation 2
Now, substitute \( R_2 \) into Equation 2:
\[ R_1 (50 - R_1) = 600 \]
Expanding this gives:
\[ 50R_1 - R_1^2 = 600 \]
Rearranging gives us a quadratic equation:
\[ R_1^2 - 50R_1 + 600 = 0 \quad \text{(Equation 3)} \]

### Step 4: Solve the Quadratic Equation
To solve the quadratic equation \( R_1^2 - 50R_1 + 600 = 0 \), we can use the quadratic formula:
\[ R_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -50 \), and \( c = 600 \):
\[ R_1 = \frac{50 \pm \sqrt{(-50)^2 - 4 \cdot 1 \cdot 600}}{2 \cdot 1} \]
Calculating the discriminant:
\[ R_1 = \frac{50 \pm \sqrt{2500 - 2400}}{2} \]
\[ R_1 = \frac{50 \pm \sqrt{100}}{2} \]
\[ R_1 = \frac{50 \pm 10}{2} \]
This gives us two possible values for \( R_1 \):
\[ R_1 = \frac{60}{2} = 30 \quad \text{or} \quad R_1 = \frac{40}{2} = 20 \]

### Step 5: Find Corresponding \( R_2 \) Values
Using \( R_1 = 30 \):
\[ R_2 = 50 - 30 = 20 \]
Using \( R_1 = 20 \):
\[ R_2 = 50 - 20 = 30 \]

### Final Answer
The individual resistances are:
\[ R_1 = 30 \, \Omega \quad \text{and} \quad R_2 = 20 \, \Omega \]