Circles

A circle is simply all the points that are the same distance from one center point, like everywhere a dog on a leash can reach at full stretch. The circle's equation looks heavy, but it has three small pieces: $h$ and $k$ (where the center is) and $r$ (the radius, the length of the leash). The distance and midpoint formulas are the tools that get you there, and the distance formula is really just the Pythagorean theorem. Bear with it; it's one idea.

The distance formula. Two points form the ends of the slanted side of a right triangle. The horizontal leg is $x_2 - x_1$ and the vertical leg is $y_2 - y_1$, so by Pythagoras the slanted side is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. The midpoint is easier still: halfway between two numbers is their average, so you average each coordinate.

Where the circle's equation comes from. A point $(x, y)$ is on the circle exactly when its distance to $(h, k)$ is $r$. Write that with the distance formula:

Squaring both sides gives the standard form. So the circle's equation isn't something new to memorize; it's the distance formula with the distance fixed.

General form. If you multiply everything out you get $x^2 + y^2 + Dx + Ey + F = 0$, which hides the center and radius. To get them back, complete the square in $x$ and in $y$.

A common mistake. In $(x + 3)^2 + (y - 2)^2 = 25$, it's tempting to read the center as $(3, -2)$, or as $(3, 2)$. It's $(-3, 2)$. The form has minus signs built in, $(x - h)$, and $x + 3$ is $x - (-3)$, so $h = -3$. Also remember the right side is $r^2$, so the radius here is $5$, not $25$.

GPS finds your position by intersecting circles (spheres, really) around satellites; collision detection in games checks whether two circles' centers are closer than the sum of their radii.