The resultant of forces P and Q acting at a point including a certain angle `alpha` is R, that of the forces 2P and Q acting at the same angle is 2R and that of P and 2Q acting at the supplementary angle is 2R. Then `P:Q:R`=

Options

A. `1:2:3`
B. `sqrt""6: sqrt""2: sqrt""5`
C. `sqrt""2: sqrt""3: sqrt""5`
D. None

Answer

To solve the problem, we need to analyze the relationships between the forces \( P \), \( Q \), and \( R \) based on the given conditions.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- The resultant of forces \( P \) and \( Q \) acting at an angle \( \alpha \) is \( R \).
- The resultant of forces \( 2P \) and \( Q \) acting at the same angle \( \alpha \) is \( 2R \).
- The resultant of forces \( P \) and \( 2Q \) acting at the supplementary angle \( (180^\circ - \alpha) \) is \( 2R \).

2. **Using the Formula for Resultant of Two Forces**:
The resultant \( R \) of two forces \( A \) and \( B \) acting at an angle \( \theta \) can be expressed as:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos(\theta)} \]

3. **Setting Up the First Equation**:
For forces \( P \) and \( Q \):
\[ R = \sqrt{P^2 + Q^2 + 2PQ \cos(\alpha)} \quad \text{(Equation 1)} \]

4. **Setting Up the Second Equation**:
For forces \( 2P \) and \( Q \):
\[ 2R = \sqrt{(2P)^2 + Q^2 + 2(2P)Q \cos(\alpha)} \]
Simplifying gives:
\[ 2R = \sqrt{4P^2 + Q^2 + 4PQ \cos(\alpha)} \quad \text{(Equation 2)} \]

5. **Squaring Both Sides of Equation 1 and Equation 2**:
- From Equation 1:
\[ R^2 = P^2 + Q^2 + 2PQ \cos(\alpha) \quad \text{(1)} \]
- From Equation 2:
\[ 4R^2 = 4P^2 + Q^2 + 4PQ \cos(\alpha) \quad \text{(2)} \]

6. **Relating the Two Equations**:
From (1), we can express \( R^2 \):
\[ 4R^2 = 4(P^2 + Q^2 + 2PQ \cos(\alpha)) = 4P^2 + 4Q^2 + 8PQ \cos(\alpha) \]
Now equate it with (2):
\[ 4P^2 + 4Q^2 + 8PQ \cos(\alpha) = 4P^2 + Q^2 + 4PQ \cos(\alpha) \]
Simplifying gives:
\[ 3Q^2 + 4PQ \cos(\alpha) = 0 \quad \text{(Equation 3)} \]

7. **Setting Up the Third Equation**:
For forces \( P \) and \( 2Q \) at angle \( (180^\circ - \alpha) \):
\[ 2R = \sqrt{P^2 + (2Q)^2 - 4PQ \cos(\alpha)} \]
Simplifying gives:
\[ 4R^2 = P^2 + 4Q^2 - 4PQ \cos(\alpha) \quad \text{(Equation 4)} \]

8. **Equating the Two Results**:
From (1):
\[ 4R^2 = 4(P^2 + Q^2 + 2PQ \cos(\alpha)) \]
Equate this with (4):
\[ 4(P^2 + Q^2 + 2PQ \cos(\alpha)) = P^2 + 4Q^2 - 4PQ \cos(\alpha) \]
This leads to:
\[ 3P^2 + 8PQ \cos(\alpha) - 3Q^2 = 0 \quad \text{(Equation 5)} \]

9. **Solving the System of Equations**:
Now we have two equations (3) and (5):
- From (3): \( 3Q^2 + 4PQ \cos(\alpha) = 0 \)
- From (5): \( 3P^2 + 8PQ \cos(\alpha) - 3Q^2 = 0 \)

Substituting \( 4PQ \cos(\alpha) = -3Q^2 \) into (5) gives:
\[ 3P^2 - 6Q^2 = 0 \implies P^2 = 2Q^2 \implies P = \sqrt{2}Q \]

10. **Finding the Ratio**:
Now substituting \( P \) back into the expression for \( R \):
\[ R^2 = P^2 + Q^2 + 2PQ \cos(\alpha) \]
Using \( P = \sqrt{2}Q \):
\[ R^2 = 2Q^2 + Q^2 + 2(\sqrt{2}Q)Q \cos(\alpha) = 3Q^2 + 2\sqrt{2}Q^2 \cos(\alpha) \]
The ratios \( P:Q:R \) can be derived from the expressions for \( P \), \( Q \), and \( R \).

### Final Ratio:
After simplifying, we find:
\[ P:Q:R = \sqrt{2}:1:1 \]