Let x,y and z are positive reals and `x ^(2) + xy + y ^(2)=2,y ^(2)+yz+z ^(2) =1 and z ^(2) +zx+x^(2) =3.` If the value of `xy + zx` can be expressed as `sqrt((p)/(q))` where p and q are relatively prime positive integral find the value of `p-q,`

Answer

To solve the problem step by step, we will analyze the given equations and derive the required expression for \(xy + zx\).

### Step 1: Analyze the given equations
We have three equations:
1. \( x^2 + xy + y^2 = 2 \) (Equation 1)
2. \( y^2 + yz + z^2 = 1 \) (Equation 2)
3. \( z^2 + zx + x^2 = 3 \) (Equation 3)

### Step 2: Rewrite the equations
We can rewrite each equation in a more useful form by completing the square.

**For Equation 1:**
\[ x^2 + xy + y^2 = 2 \implies x^2 + 2xy + y^2 = 2 + xy \implies (x + y)^2 = 2 + xy \]
Let’s denote this as Equation 4.

**For Equation 2:**
\[ y^2 + yz + z^2 = 1 \implies y^2 + 2yz + z^2 = 1 + yz \implies (y + z)^2 = 1 + yz \]
Let’s denote this as Equation 5.

**For Equation 3:**
\[ z^2 + zx + x^2 = 3 \implies z^2 + 2zx + x^2 = 3 + zx \implies (z + x)^2 = 3 + zx \]
Let’s denote this as Equation 6.

### Step 3: Combine the equations
Now, we add Equations 4, 5, and 6:
\[ (x + y)^2 + (y + z)^2 + (z + x)^2 = (2 + xy) + (1 + yz) + (3 + zx) \]
This simplifies to:
\[ (x + y)^2 + (y + z)^2 + (z + x)^2 = 6 + xy + yz + zx \]
Let’s denote this as Equation 7.

### Step 4: Use the AM-GM inequality
Using the Arithmetic Mean-Geometric Mean (AM-GM) inequality, we have:
\[ \frac{x + y}{2} \geq \sqrt{xy}, \quad \frac{y + z}{2} \geq \sqrt{yz}, \quad \frac{z + x}{2} \geq \sqrt{zx} \]
Squaring and adding these inequalities gives:
\[ \frac{(x + y)^2 + (y + z)^2 + (z + x)^2}{4} \geq xy + yz + zx \]
Substituting from Equation 7:
\[ \frac{6 + xy + yz + zx}{4} \geq xy + yz + zx \]

### Step 5: Solve the inequality
Let \( T = xy + yz + zx \). Then we have:
\[ \frac{6 + T}{4} \geq T \]
Multiplying through by 4:
\[ 6 + T \geq 4T \implies 6 \geq 3T \implies T \leq 2 \]

### Step 6: Check for equality
To find the maximum value of \( T \), we assume equality in the AM-GM inequality:
\[ \frac{x + y}{2} = \sqrt{xy}, \quad \frac{y + z}{2} = \sqrt{yz}, \quad \frac{z + x}{2} = \sqrt{zx} \]
This leads us to conclude that:
\[ xy + yz + zx = 2 \]

### Step 7: Express \( T \) in the required form
We need to express \( T \) in the form \( \sqrt{\frac{p}{q}} \):
\[ T = 2 = \sqrt{\frac{4}{1}} \]
Here, \( p = 4 \) and \( q = 1 \).

### Step 8: Find \( p - q \)
Finally, we compute:
\[ p - q = 4 - 1 = 3 \]

### Final Answer
The value of \( p - q \) is \( \boxed{3} \).