Circles, Distance, and Midpoints

Definition The distance between the points $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$ in the coordinate plane is

It comes from the Pythagorean theorem applied to the right triangle whose legs have lengths $\lvert x_2 - x_1 \rvert$ and $\lvert y_2 - y_1 \rvert$ and whose hypotenuse is the segment from $P_1$ to $P_2$. When the two points lie on one horizontal or vertical line there is no triangle, and the formula gives the plain gap $\lvert x_2 - x_1 \rvert$ or $\lvert y_2 - y_1 \rvert$, as it should. The order of the points does not matter, because squaring removes the signs.

Definition The midpoint of the segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is

It lies on the segment, and its distance to each endpoint is half the length of the segment.

Definition Let $C$ be a point of the plane and $r$ a positive real number. The circle with center $C$ and radius $r$ is the set of all points $P$ in the plane with $d(P, C) = r$. The word radius also names any segment from the center to a point of the circle. A diameter is a segment through the center with both endpoints on the circle. Every diameter has length $2r$, and the center is the midpoint of every diameter. The center itself is not a point of the circle.

Definition A point $(x, y)$ lies on the circle with center $(h, k)$ and radius $r > 0$ exactly when

This is the standard equation of the circle. It is the distance formula with both sides squared, which loses nothing because a distance and $r$ are both nonnegative. Points with $(x - h)^2 + (y - k)^2 < r^2$ lie inside the circle and points with $(x - h)^2 + (y - k)^2 > r^2$ lie outside. The unit circle $x^2 + y^2 = 1$ has center $(0, 0)$ and radius $1$.

Definition Expanding a standard equation and moving everything to one side gives an equation of the form

with real numbers $D$, $E$ and $F$, called the general equation of the circle. Conversely, every equation of this form describes one of three sets.

• When $D^2 + E^2 - 4F > 0$ it is the circle with center $\left(-\frac{D}{2}, -\frac{E}{2}\right)$ and radius $\frac{1}{2}\sqrt{D^2 + E^2 - 4F}$.
• When $D^2 + E^2 - 4F = 0$ it is the single point $\left(-\frac{D}{2}, -\frac{E}{2}\right)$.
• When $D^2 + E^2 - 4F < 0$ no point of the plane satisfies it.

These formulas come from completing the square, as the next section shows. An equation $Ax^2 + Ay^2 + Dx + Ey + F = 0$ with $A \ne 0$ can first be divided by $A$. In this form $x^2$ and $y^2$ have the same coefficient and there is no $xy$ term, which is how you recognize a possible circle.

Definition To find the center and radius from a general equation, group the $x$ terms and the $y$ terms, move the constant to the other side, and complete the square in each variable with

adding the same amounts to both sides. The result has the form $(x - h)^2 + (y - k)^2 = s$. It is a circle exactly when $s > 0$, and then its radius is $\sqrt{s}$.