The resultant of tow forces P and Q is H. If Q is doubled, R is doubled and if Q is reversed, R is again doubled. If the ratio `P^(2): Q^(2): R^(2)= 2: 3: x` then x is equal to

Options

A. 5
B. 4
C. 3
D. 2

Answer

To solve the problem step by step, we will analyze the given conditions and derive the required value of \( x \).

### Step 1: Understand the Given Forces and Resultants
We have two forces \( P \) and \( Q \) whose resultant is \( H \). The relationship can be expressed using the formula for the resultant of two forces:
\[ H^2 = P^2 + Q^2 + 2PQ \cos(\theta) \]
where \( \theta \) is the angle between the forces \( P \) and \( Q \).

### Step 2: Condition When \( Q \) is Doubled
When \( Q \) is doubled, the new resultant \( R \) becomes:
\[ R^2 = P^2 + (2Q)^2 + 2P(2Q) \cos(\theta) \]
This simplifies to:
\[ R^2 = P^2 + 4Q^2 + 4PQ \cos(\theta) \]

### Step 3: Condition When \( Q \) is Reversed
When \( Q \) is reversed, the resultant becomes:
\[ R^2 = P^2 + (-Q)^2 + 2P(-Q) \cos(\theta) \]
This simplifies to:
\[ R^2 = P^2 + Q^2 - 2PQ \cos(\theta) \]

### Step 4: Set Up the Equations
From the above conditions, we have two equations for \( R^2 \):
1. \( R^2 = P^2 + 4Q^2 + 4PQ \cos(\theta) \) (when \( Q \) is doubled)
2. \( R^2 = P^2 + Q^2 - 2PQ \cos(\theta) \) (when \( Q \) is reversed)

### Step 5: Equate the Two Expressions for \( R^2 \)
Setting the two expressions for \( R^2 \) equal gives:
\[ P^2 + 4Q^2 + 4PQ \cos(\theta) = P^2 + Q^2 - 2PQ \cos(\theta) \]
This simplifies to:
\[ 4Q^2 + 4PQ \cos(\theta) = Q^2 - 2PQ \cos(\theta) \]
Rearranging gives:
\[ 3Q^2 + 6PQ \cos(\theta) = 0 \]
From this, we can express \( \cos(\theta) \):
\[ \cos(\theta) = -\frac{Q^2}{2PQ} \]

### Step 6: Substitute Back to Find \( R^2 \)
Now we can substitute \( \cos(\theta) \) back into one of the equations for \( R^2 \):
\[ R^2 = P^2 + Q^2 - 2PQ \left(-\frac{Q^2}{2PQ}\right) \]
This simplifies to:
\[ R^2 = P^2 + Q^2 + Q^2 = P^2 + 2Q^2 \]

### Step 7: Establish the Ratio
We know that the ratio \( P^2 : Q^2 : R^2 = 2 : 3 : x \). From our derived equations:
- Let \( P^2 = 2k \)
- Let \( Q^2 = 3k \)
- Then \( R^2 = P^2 + 2Q^2 = 2k + 2(3k) = 2k + 6k = 8k \)

Thus, we have:
\[ R^2 = 8k \]

### Step 8: Find \( x \)
Now, we can relate this back to the ratio:
\[ P^2 : Q^2 : R^2 = 2k : 3k : 8k \]
From this, we can see that:
\[ x = 8 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{8} \]