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.