Find the total number of integral solutions (a, b) of (7-ab)^2/a^2 + b^2
=1

We are tasked with finding the total number of integral solutions \((a, b)\) for the equation:

\[
\frac{(7 - ab)^2}{a^2 + b^2} = 1
\]

### Step 1: Rearranging the equation
We can start by multiplying both sides of the equation by \( a^2 + b^2 \) to eliminate the denominator:

\[
(7 - ab)^2 = a^2 + b^2
\]

This equation implies that the square of \( 7 - ab \) equals the sum of squares of \( a \) and \( b \).

### Step 2: Exploring possible values of \( a \) and \( b \)
We now need to find integer values of \( a \) and \( b \) such that the equation holds.

To do this, we can examine specific values of \( a \) and \( b \), and check if they satisfy the equation. Let's explore some values systematically.

### Step 3: Calculation of integer solutions
We will test small integer values of \( a \) and \( b \) to determine when the equation is satisfied.

Let me now compute the possible values for \( a \) and \( b \).There are 8 integral solutions \((a, b)\) that satisfy the given equation:

\[
(-7, 0), (-4, -3), (-3, -4), (0, -7), (0, 7), (3, 4), (4, 3), (7, 0)
\]

### Summary:
- The total number of integral solutions is **8**.

Would you like further details or explanations on this problem? Here are five questions to consider:

1. How can we generalize this approach to more complex equations?
2. Why did we restrict \(a\) and \(b\) to the range \([-10, 10]\)?
3. How does the symmetry in the equation help reduce the number of calculations?
4. Can this equation have non-integer solutions as well?
5. What happens if we modify the constant 7 to another value?

**Tip**: When solving diophantine equations, testing small integer values often helps identify patterns or solutions quickly.

Find the total number of integral solutions (a, b) of (7 - ab)^2 / (a^2 + b^2) = 1

This detailed solution finds 8 integral solutions for the equation (7 - ab)^2 / (a^2 + b^2) = 1. Key concepts include algebra, symmetry, and diophantine equations. Suitable for Grades 10-12 students exploring sum of squares problems.

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