The equation of a circle is given as $(x - 5)^2 + (y - 1)^2 = 9$. We want to determine if the point $(8, 2)$ is on the circle, inside the circle, or outside the circle.

1. Problem Description

The equation of a circle is given as $(x - 5)^2 + (y - 1)^2 = 9$. We want to determine if the point $(8, 2)$ is on the circle, inside the circle, or outside the circle.

2. Solution Steps

The general equation of a circle with center $(h, k)$ and radius $r$ is given by:

$(x - h)^2 + (y - k)^2 = r^2$

In our case, the circle has center $(5, 1)$ and radius $r$ such that $r^2 = 9$, so $r = 3$.

To determine the position of the point $(8, 2)$ relative to the circle, we can plug the point's coordinates into the left side of the circle's equation and compare the result to the right side (which is $r^2 = 9$).

Plugging in $(x, y) = (8, 2)$ into the equation, we get:

$(8 - 5)^2 + (2 - 1)^2 = (3)^2 + (1)^2 = 9 + 1 = 10$

Now we compare the result to $r^2 = 9$:

Since $10 > 9$, the point $(8, 2)$ is outside the circle.

3. Final Answer

The point (8, 2) is outside the circle.