The number of non-negative integral ordered pair (s) `(x,y) ` for which `(xy-7)^(2) =x ^(2) + y^(2)` holds is greater then or equal to :

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the equation \((xy - 7)^2 = x^2 + y^2\) for non-negative integral ordered pairs \((x, y)\), we will follow these steps:

### Step 1: Expand the equation
Start by expanding the left-hand side of the equation:
\[ (xy - 7)^2 = x^2 + y^2 \]
Expanding the left side gives:
\[ x^2y^2 - 14xy + 49 = x^2 + y^2 \]

### Step 2: Rearrange the equation
Rearranging the equation to bring all terms to one side:
\[ x^2y^2 - 14xy + 49 - x^2 - y^2 = 0 \]
This simplifies to:
\[ x^2y^2 - x^2 - y^2 - 14xy + 49 = 0 \]

### Step 3: Factor the equation
To make it easier to analyze, we can rearrange it further:
\[ x^2y^2 - 14xy + 49 = x^2 + y^2 \]
This can be rewritten as:
\[ x^2y^2 - 14xy + 49 - (x^2 + y^2) = 0 \]

### Step 4: Consider the structure of the equation
Notice that the left side can be factored or analyzed for specific values. We can rewrite it as:
\[ (xy - 7)^2 = x^2 + y^2 \]
This indicates that we can analyze the values of \(xy\) and \(x^2 + y^2\).

### Step 5: Set up a system of equations
From the equation, we can derive:
\[ xy - 7 = \pm \sqrt{x^2 + y^2} \]
This gives us two cases to consider:
1. \(xy - 7 = \sqrt{x^2 + y^2}\)
2. \(xy - 7 = -\sqrt{x^2 + y^2}\)

### Step 6: Analyze the first case
For the first case:
\[ xy - 7 = \sqrt{x^2 + y^2} \]
Squaring both sides gives:
\[ (xy - 7)^2 = x^2 + y^2 \]
This leads us back to our original equation, so we need to find specific values for \(x\) and \(y\).

### Step 7: Analyze the second case
For the second case:
\[ xy - 7 = -\sqrt{x^2 + y^2} \]
This implies:
\[ xy + \sqrt{x^2 + y^2} = 7 \]
Squaring both sides again leads to a similar analysis.

### Step 8: Solve for integer pairs
To find non-negative integral solutions, we can test small values of \(x\) and \(y\):
- If \(x = 0\), then \(y = 7\).
- If \(y = 0\), then \(x = 7\).
- If \(x = 1\), then \(y\) can be calculated.
- Continue this process for \(x = 2, 3, 4, 5, 6, 7\).

### Step 9: Count the solutions
By testing pairs, we find the valid combinations:
- \((0, 7)\)
- \((1, 6)\)
- \((2, 5)\)
- \((3, 4)\)
- \((4, 3)\)
- \((5, 2)\)
- \((6, 1)\)
- \((7, 0)\)

This gives us a total of 8 non-negative integral ordered pairs.

### Conclusion
Thus, the number of non-negative integral ordered pairs \((x, y)\) for which \((xy - 7)^2 = x^2 + y^2\) holds is greater than or equal to 8.