Conic Sections/Circle

Definition

The circle is the simplest and best known conic section. As a conic section, the circle is the intersection of a plane perpendicular to the cone's axis.

The geometric definition of a circle is the locus of all points a constant distance $r$ from a point $(h,k)$ . The distance $r$ is the radius of the circle, and the point $(h,k)$ is the circle's center.

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Equations

General Form

The general equation for a circle with center $(h,k)$ and radius $r$ is

$(x-h)^2+(y-k)^2=r^2$ .

In the simplest case of a circle whose center is at the origin, the equation is simply a restatement of the Pythagorean Theorem:

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

General form

The general form of a circle equation is

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

<-g,-f> is the center of the circle.

Polar Coordinates

In the case of a circle centered at the origin, the polar equation of a circle is very simple because polar coordinates are essentially based on circles. For a circle with radius $a$ ,

$r = a$ .

In the more complicated case of a circle with an arbitrary location, the equation is

$r^2 - 2 r r_0 \cos(\theta - \varphi) + r_0^2 = a^2$ ,
where $r_0$ is the distance from the circle's center to the origin and $\varphi$ is the angle pointing to the circle.

There are many cases that allow the equation to be simplified. If a point on the circle is touching the origin, its polar equation may consist of a single trig function.

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