1. **State the problem:** Given points $P=(6,5)$ and $Q=(2,1)$ as endpoints of the diameter of a circle, find the equation of the circle. 2. **Formula and rules:** The equation of a circle with center $(h,k)$ and radius $r$ is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ The center $(h,k)$ is the midpoint of the diameter endpoints. The radius $r$ is half the distance between $P$ and $Q$. 3. **Find the center:** Midpoint formula: $$ h = \frac{x_1 + x_2}{2} = \frac{6 + 2}{2} = 4 $$ $$ k = \frac{y_1 + y_2}{2} = \frac{5 + 1}{2} = 3 $$ So, center is $(4,3)$. 4. **Find the radius:** Distance between $P$ and $Q$: $$ d = \sqrt{(6 - 2)^2 + (5 - 1)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} $$ Radius $r$ is half of $d$: $$ r = \frac{4\sqrt{2}}{2} = 2\sqrt{2} $$ 5. **Write the equation:** $$ (x - 4)^2 + (y - 3)^2 = (2\sqrt{2})^2 = 8 $$ **Final answer:** $$(x - 4)^2 + (y - 3)^2 = 8$$