Circle Equations, Centers and Radii

Definition The distance between the points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is

It is the Pythagorean theorem applied to the right triangle whose legs are the horizontal change $x_2 - x_1$ and the vertical change $y_2 - y_1$. Neither this formula nor the circle equation is on the SAT reference sheet, so both are worth knowing by heart.

Definition A circle is the set of all points in a plane at a fixed positive distance $r$ from a fixed point. The fixed point is the center and the distance $r$ is the radius. A segment from the center to a point of the circle is also called a radius.

Definition A point $(x, y)$ lies on the circle with center $(h, k)$ and radius $r > 0$ exactly when its distance from $(h, k)$ is $r$. Squaring both sides of that distance condition gives the standard form

The constant on the right is the radius squared, not the radius. Signs inside the parentheses are subtractions, so $(x + 4)^2$ means $h = -4$. To see a true circle on a graph, the axes must use the same scale, since unequal scales stretch it into an oval.

Definition To write the equation of a circle from its center $(h, k)$ and one point $(x_1, y_1)$ on it, compute

and substitute into standard form. Moving a circle $a$ units right and $b$ units up moves its center to $(h + a, k + b)$ and leaves the radius unchanged, so the right side $r^2$ stays the same.

Definition Expanding standard form gives an equation like $x^2 + y^2 + Dx + Ey + F = 0$. To return to standard form, complete the square separately in $x$ and in $y$, using

and the same identity for $y$. Whatever you add to one side you must add to the other. The result has the form $(x - h)^2 + (y - k)^2 = c$. If $c > 0$, the graph is a circle with radius $\sqrt{c}$. If $c = 0$, the only solution is the single point $(h, k)$. If $c < 0$, no real point satisfies the equation, because a sum of squares of real numbers is never negative.

Definition An equation $ax^2 + ay^2 + Dx + Ey + F = 0$ in which $x^2$ and $y^2$ have the same nonzero coefficient $a$ can be normalized by dividing every term by $a$. This gives an equivalent equation with leading coefficients $1$, which is ready for completing the square. If the coefficients of $x^2$ and $y^2$ are different, the graph is not a circle.

Definition A diameter is a chord through the center, and its length is $2r$. If $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints of a diameter, the center is their midpoint

and the radius is half the distance between them.