Show that the set all points such that the difference of their distances from `(4,0)a n d(-4,0)` is always equal to 2 represents a hyperbola.

Explanation

To show that the set of all points such that the difference of their distances from the points \( (4, 0) \) and \( (-4, 0) \) is always equal to 2 represents a hyperbola, we can follow these steps:

### Step 1: Define the points and the distances
Let \( A(4, 0) \) and \( B(-4, 0) \) be the two fixed points. Let \( P(x, y) \) be a point in the plane. The distances from \( P \) to \( A \) and \( B \) are given by:
\[ PA = \sqrt{(x - 4)^2 + y^2} \]
\[
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