Prove the equation `sqrt((x + 4)^(2) + (y + 2)^(2)) - sqrt((x-4)^(2) + (y - 2)^(2)) = 8` represents a hyperbola.

Answer

To prove that the equation

\[ \sqrt{(x + 4)^{2} + (y + 2)^{2}} - \sqrt{(x - 4)^{2} + (y - 2)^{2}} = 8 \]

represents a hyperbola, we will follow these steps:

### Step 1: Identify the distances
Let \( S_1 \) be the distance from the point \( (x, y) \) to the point \( (-4, -2) \) and \( S_2 \) be the distance from the point \( (x, y) \) to the point \( (4, 2) \).

Thus, we can express \( S_1 \) and \( S_2 \) as:
\[ S_1 = \sqrt{(x + 4)^{2} + (y + 2)^{2}} \]
\[ S_2 = \sqrt{(x - 4)^{2} + (y - 2)^{2}} \]

### Step 2: Rewrite the equation
The given equation can be rewritten as:
\[ S_1 - S_2 = 8 \]

### Step 3: Understand the definition of a hyperbola
A hyperbola is defined as the set of points \( (x, y) \) such that the absolute difference of the distances to two fixed points (foci) is a constant. In our case, the fixed points are \( (-4, -2) \) and \( (4, 2) \).

### Step 4: Compare with the hyperbola definition
From the definition, we can see that:
\[ |S_1 - S_2| = 8 \]
This indicates that the absolute difference of the distances from the point \( (x, y) \) to the two foci is a constant (8).

### Step 5: Identify the foci and the constant
The foci of the hyperbola are:
- \( F_1 = (-4, -2) \)
- \( F_2 = (4, 2) \)

The constant \( 2a \) in the hyperbola equation is equal to 8, which gives us \( a = 4 \).

### Step 6: Conclusion
Since the equation \( |S_1 - S_2| = 8 \) fits the definition of a hyperbola, we conclude that the given equation represents a hyperbola.