1. **Problem Statement:** Find the standard form of the equation of a circle given different conditions. 2. **Formula:** The standard form of the equation of a circle with center at $(h,k)$ and radius $r$ is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ 3. **Example 1:** Center at $(2,1)$ and radius $2$. Substitute $h=2$, $k=1$, and $r=2$: $$ (x - 2)^2 + (y - 1)^2 = 2^2 $$ $$ (x - 2)^2 + (y - 1)^2 = 4 $$ This is the standard form. 4. **Example 2:** Center at $(-1,1)$ and diameter $4$. Radius $r$ is half the diameter: $$ r = \frac{4}{2} = 2 $$ Substitute $h=-1$, $k=1$, and $r=2$: $$ (x + 1)^2 + (y - 1)^2 = 2^2 $$ $$ (x + 1)^2 + (y - 1)^2 = 4 $$ 5. **Example 3:** Center at $(1,0)$ and passes through $(-1,-1)$. First, find radius $r$ by calculating the distance between center and point: $$ r = \sqrt{(-1 - 1)^2 + (-1 - 0)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} $$ Substitute $h=1$, $k=0$, and $r=\sqrt{5}$: $$ (x - 1)^2 + y^2 = (\sqrt{5})^2 $$ $$ (x - 1)^2 + y^2 = 5 $$ **Final answers:** 1. $$(x - 2)^2 + (y - 1)^2 = 4$$ 2. $$(x + 1)^2 + (y - 1)^2 = 4$$ 3. $$(x - 1)^2 + y^2 = 5$$ These equations represent circles with the given centers and radii.