The Circle

1. Core Concept Overview & Intuition

A circle is defined as the locus of a point which moves such that its distance from a fixed point is constant. The fixed point is called the center of the circle, and the constant distance from the center is called the radius. This definition is the foundation of every result in this unit: every equation of a circle is simply an algebraic encoding of the statement "the distance from a variable point to a fixed point equals a fixed length."

The intuition behind the standard equation is direct. If a point $(x,y)$ lies on a circle with center $(a,b)$ and radius $r$, then the distance between $(x,y)$ and $(a,b)$ must equal $r$. Expressing that distance condition algebraically produces the standard equation. When the center is placed at the origin, the equation simplifies considerably, since the fixed point is $(0,0)$.

Expanding the standard equation and collecting terms produces the general equation of a circle. Conversely, when a circle is presented in general form, the center and radius can be recovered by completing the square. This two-way relationship between the standard and general forms is the central technical skill of the topic.

Beyond the equation itself, the unit studies the geometry of a circle in relation to lines and external points. A tangent touches the circle at exactly one point, and a normal is perpendicular to the tangent at the point of contact. From an external point, two tangents can be drawn to a circle, and the chord joining the two points of contact is called the chord of contact. The distance from an external point to a circle is measured along the line joining the point to the center. Finally, the intersection of a line and a circle is determined by solving their equations simultaneously, and the number of solutions corresponds to the number of intersection points.

2. Mathematical Formalization & Core Definitions

2.1 Standard Equation of a Circle

Consider a circle with center $(a,b)$ and radius $r$. Let $(x,y)$ be any point on the circle. By the definition of the circle, the distance from $(x,y)$ to $(a,b)$ equals $r$. Hence the equation of the circle is

$$
(x-a)^2+(y-b)^2=r^2.
$$

If the center of the circle is at the origin, so that $(a,b)=(0,0)$, the equation becomes

$$
x^2+y^2=r^2.
$$

Here $x$ and $y$ are the coordinates of a general point on the circle, $(a,b)$ is the fixed center, and $r$ is the radius, with $r>0$.

2.2 General Equation of a Circle

Expanding the standard equation gives

$$
x^2+y^2-2ax-2by+a^2+b^2-r^2=0.
$$

This can be written in the form

$$
x^2+y^2+2gx+2fy+c=0,
$$

where $g=-a$, $f=-b$, and $c=a^2+b^2-r^2$. This is called the general equation of the circle.

If the equation of a circle is given in general form, the center and the radius can be obtained by completing the squares. Completing the square yields

$$
(x+g)^2+(y+f)^2=g^2+f^2-c.
$$

This is the equation of a circle with center $(-g,-f)$ and radius

$$
r=\sqrt{g^2+f^2-c}.
$$

The quantity $g^2+f^2-c$ governs the nature of the circle:

• If $g^2+f^2-c=0$, then the equation represents a point which coincides with the center $(-g,-f)$.
• If $g^2+f^2-c<0$, then the radius is imaginary, and the circle is referred to as a virtual circle.
• If $g^2+f^2-c>0$, then the circle is real.

2.3 Distance from a Point to a Circle

Consider a point $P$ outside the circle. Let $Q$ be a point on the circle such that $Q$ lies on the line segment $PC$, where $C$ is the center of the circle. The distance from $P$ to the circle is the length $PQ$, which equals $PC-r$.

In general, the distance $d$ from a point to a circle is given by

$$
d=PC-r,
$$

with $d$ positive if the point is outside the circle and negative if the point is inside the circle. Here $PC$ denotes the distance from the point to the center and $r$ is the radius.

2.4 Tangents, Normals, and the Chord of Contact

A tangent to a circle is a line that touches the circle at exactly one point, called the point of contact. A normal to a circle at a point of contact is the line perpendicular to the tangent at that point; for a circle, the normal at a point passes through the center.

From an external point, two tangents can be drawn to a circle. If the two points of contact are $A$ and $B$, the chord through $A$ and $B$ is called the chord of contact of tangents from the external point with respect to the given circle.