1. **State the problem:** We need to find the equation of a circle centered at the point $(5, 1)$ with a radius of $2$ units. 2. **Recall the formula for a circle's equation:** The standard form of a circle's equation with center $(h, k)$ and radius $r$ is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ 3. **Identify the values:** Here, $h = 5$, $k = 1$, and $r = 2$. 4. **Substitute the values into the formula:** $$ (x - 5)^2 + (y - 1)^2 = 2^2 $$ 5. **Simplify the radius squared:** $$ (x - 5)^2 + (y - 1)^2 = 4 $$ 6. **Final answer:** The equation of the circle is $$ (x - 5)^2 + (y - 1)^2 = 4 $$ This equation represents all points $(x, y)$ that are exactly $2$ units away from the center $(5, 1)$ on the coordinate plane.