Circle equations

Every circle in the plane can be written as $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the centre and $r$ is the radius. The centre coordinates are subtracted inside the brackets, and the number on the right is the radius squared.

Read the graph above. The equation $(x - 3)^2 + (y + 4)^2 = 25$ matches the form with $h = 3$ and $k = -4$, because $(y + 4)$ is really $(y - (-4))$. And the right side $25$ is $r^2$, so $r = 5$, not $25$. Watch those two things: the sign inside the brackets, and the square root of the right side.

To go the other way and build the equation, put the centre and radius into the form.
For centre $(4, -2)$ and radius $5$: $(x - 4)^2 + (y - (-2))^2 = 5^2$, which is $(x - 4)^2 + (y + 2)^2 = 25$.

Find the centre and radius of $(x - 5)^2 + (y - 2)^2 = 36$.
Compare with $(x - h)^2 + (y - k)^2 = r^2$: $h = 5$, $k = 2$, and $r^2 = 36$ so $r = 6$.
Centre $(5, 2)$, radius $6$.

Find the centre and radius of $(x + 1)^2 + (y - 3)^2 = 49$.
Rewrite $(x + 1)$ as $(x - (-1))$, so $h = -1$ and $k = 3$.
And $r^2 = 49$ gives $r = 7$.
Centre $(-1, 3)$, radius $7$.

Write the equation of the circle with centre $(2, -6)$ and radius $4$.
Substitute into $(x - h)^2 + (y - k)^2 = r^2$:
$(x - 2)^2 + (y + 6)^2 = 16$.