Two concurrent forces are of magnitudes (P+Q) newtons and (P-Q) newtons. Then angle between their line of acting is `135^(@)`, and their resultant is a force of 2 newtons, perpendicular to the line of action of the second force. Then the values of P and Q are respectively.

Options

A. `(sqrt""2+1)N, (sqrt""2-1)N`
B. `(sqrt""2-1)N, (sqrt""+1)N`
C. `(sqrt""3+1)N, (sqrt""-1)N`
D. `(sqrt""3-1)N, (sqrt""3+1)N`

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the Forces and Angle
We have two forces:
1. \( F_1 = P + Q \) newtons
2. \( F_2 = P - Q \) newtons

The angle between these two forces is \( 135^\circ \).

### Step 2: Use the Law of Cosines to Find the Resultant
The resultant \( R \) of two forces can be calculated using the formula:
\[ R^2 = F_1^2 + F_2^2 - 2 F_1 F_2 \cos(\theta) \]
Substituting the values:
\[ R^2 = (P + Q)^2 + (P - Q)^2 - 2(P + Q)(P - Q) \cos(135^\circ) \]
Since \( R = 2 \) newtons, we have:
\[ 4 = (P + Q)^2 + (P - Q)^2 - 2(P + Q)(P - Q)(-\frac{1}{\sqrt{2}}) \]

### Step 3: Simplify the Equation
Expanding the squares:
\[ (P + Q)^2 = P^2 + 2PQ + Q^2 \]
\[ (P - Q)^2 = P^2 - 2PQ + Q^2 \]
Adding these gives:
\[ (P + Q)^2 + (P - Q)^2 = 2P^2 + 2Q^2 \]
Now substituting back:
\[ 4 = 2P^2 + 2Q^2 + \frac{2(P^2 - Q^2)}{\sqrt{2}} \]
This simplifies to:
\[ 4 = 2P^2 + 2Q^2 + \frac{2(P^2 - Q^2)}{\sqrt{2}} \]

### Step 4: Rearranging the Equation
Rearranging gives:
\[ 4 = 2P^2 + 2Q^2 + \frac{2P^2}{\sqrt{2}} - \frac{2Q^2}{\sqrt{2}} \]
Combining terms:
\[ 4 = 2P^2(1 + \frac{1}{\sqrt{2}}) + 2Q^2(1 - \frac{1}{\sqrt{2}}) \]

### Step 5: Solve for P and Q
Let’s denote:
\[ a = 1 + \frac{1}{\sqrt{2}}, \quad b = 1 - \frac{1}{\sqrt{2}} \]
Then we can write:
\[ 4 = 2P^2a + 2Q^2b \]
Dividing through by 2:
\[ 2 = P^2a + Q^2b \]

### Step 6: Find the Values of P and Q
We need another equation to solve for \( P \) and \( Q \). We can use the relationship:
\[ P + Q = x \quad \text{and} \quad P - Q = y \]
From the earlier equations, we can derive \( P \) and \( Q \) in terms of \( x \) and \( y \):
\[ P = \frac{x + y}{2}, \quad Q = \frac{x - y}{2} \]
Substituting these into our equation will give us a system of equations to solve for \( P \) and \( Q \).

### Final Step: Solve the System
Using the equations derived, we can solve for \( P \) and \( Q \):
1. Substitute values back into the equations.
2. Solve the resulting quadratic equations.