The resultant of two forces P and Q is R. If Q is doubled, R is also doubled and if Q is reversed, R is again doubled. Then `P^(2): Q^(2) : R^(2)` is given by

Options

A. `1:1:1`
B. `2:2:3`
C. `2:3:2`
D. `3:2:2`

Answer

To solve the problem, we need to analyze the conditions given in the question step by step.

### Step-by-Step Solution:

1. **Understanding the Resultant Force**:
- The resultant \( R \) of two forces \( P \) and \( Q \) can be expressed using the formula:
\[ R^2 = P^2 + Q^2 + 2PQ \cos \theta \]
where \( \theta \) is the angle between the forces \( P \) and \( Q \).

2. **Condition 1: Doubling \( Q \)**:
- If \( Q \) is doubled, the new resultant \( R' \) is also doubled:
\[ R'^2 = P^2 + (2Q)^2 + 2P(2Q) \cos \theta \]
- Since \( R' = 2R \), we can write:
\[ (2R)^2 = P^2 + 4Q^2 + 4PQ \cos \theta \]
- This simplifies to:
\[ 4R^2 = P^2 + 4Q^2 + 4PQ \cos \theta \quad \text{(Equation 1)} \]

3. **Condition 2: Reversing \( Q \)**:
- If \( Q \) is reversed, the angle becomes \( 180^\circ - \theta \), and the resultant is again doubled:
\[ R'^2 = P^2 + Q^2 + 2P(-Q) \cos \theta \]
- This gives us:
\[ (2R)^2 = P^2 + Q^2 - 2PQ \cos \theta \]
- Simplifying, we have:
\[ 4R^2 = P^2 + Q^2 - 2PQ \cos \theta \quad \text{(Equation 2)} \]

4. **Equating the Two Conditions**:
- From Equation 1 and Equation 2, we can set them equal to each other:
\[ P^2 + 4Q^2 + 4PQ \cos \theta = P^2 + Q^2 - 2PQ \cos \theta \]
- Cancel \( P^2 \) from both sides:
\[ 4Q^2 + 4PQ \cos \theta = Q^2 - 2PQ \cos \theta \]
- Rearranging gives:
\[ 4Q^2 - Q^2 + 4PQ \cos \theta + 2PQ \cos \theta = 0 \]
- This simplifies to:
\[ 3Q^2 + 6PQ \cos \theta = 0 \]
- From this, we can express \( Q^2 \):
\[ Q^2 = -2PQ \cos \theta \]

5. **Substituting Back**:
- Now substitute \( Q^2 \) back into the original resultant equation:
\[ R^2 = P^2 + Q^2 + 2PQ \cos \theta \]
- Replace \( Q^2 \):
\[ R^2 = P^2 - 2PQ \cos \theta + 2PQ \cos \theta \]
- This simplifies to:
\[ R^2 = P^2 \]

6. **Finding Ratios**:
- We have \( R^2 = P^2 \) and \( Q^2 = -2PQ \cos \theta \).
- To find the ratios \( P^2 : Q^2 : R^2 \), we can express \( Q^2 \) in terms of \( R^2 \):
\[ Q^2 = 3R^2/2 \]
- Thus, the ratios become:
\[ P^2 : Q^2 : R^2 = R^2 : \frac{3}{2}R^2 : R^2 = 1 : \frac{3}{2} : 1 \]
- This can be simplified to:
\[ 2 : 3 : 2 \]

### Final Result:
The ratio \( P^2 : Q^2 : R^2 \) is \( 2 : 3 : 2 \).