Two forces are such that the sum of their magnitudes is 18N and their resultant is 12 N which is perpendicular to the smaller force. Then the magnitude of the forces are

Options

A. (a) 12N, 6N
B. (b) 13N, 5N
C. (c) 10N, 8N
D. (d) 16N, 2N

Explanation

To solve the problem, we need to find the magnitudes of two forces \( F_1 \) and \( F_2 \) given that:

1. The sum of their magnitudes is \( F_1 + F_2 = 18 \, \text{N} \).
2. The resultant of the two forces is \( R = 12 \, \text{N} \), and it is perpendicular to the smaller force \( F_2 \).

### Step 1: Set up the equations
We have two equations based on the problem statement:
1. \( F_1 + F_2 = 18 \) (Equation 1)
2. \( F_1^2 - F_2^2 = R^2 \) (Equation 2)

### Step 2: Substitute the value of the resultant
From Equation 2, we know that:
\[ R = 12 \, \text{N} \]
Thus:
\[ F_1^2 - F_2^2 = 12^2 = 144 \quad \text{(Equation 2)} \]

### Step 3: Express \( F_1 \) in terms of \( F_2 \)
From Equation 1, we can express \( F_1 \) in terms of \( F_2 \):
\[ F_1 = 18 - F_2 \quad \text{(Equation 3)} \]

### Step 4: Substitute Equation 3 into Equation 2
Now, substitute Equation 3 into Equation 2:
\[ (18 - F_2)^2 - F_2^2 = 144 \]

### Step 5: Expand and simplify
Expanding the left-hand side:
\[ (18 - F_2)(18 - F_2) - F_2^2 = 144 \]
\[ 324 - 36F_2 + F_2^2 - F_2^2 = 144 \]
This simplifies to:
\[ 324 - 36F_2 = 144 \]

### Step 6: Solve for \( F_2 \)
Rearranging gives:
\[ 36F_2 = 324 - 144 \]
\[ 36F_2 = 180 \]
\[ F_2 = \frac{180}{36} = 5 \, \text{N} \]

### Step 7: Find \( F_1 \)
Now substitute \( F_2 \) back into Equation 3 to find \( F_1 \):
\[ F_1 = 18 - F_2 = 18 - 5 = 13 \, \text{N} \]

### Conclusion
The magnitudes of the forces are:
\[ F_1 = 13 \, \text{N}, \quad F_2 = 5 \, \text{N} \]