# Conic Sections/Circle

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 ${\displaystyle r}$ from a point ${\displaystyle (h,k)}$ and forming the circumference (C). The distance ${\displaystyle r}$ is the radius (R) of the circle, and the point ${\displaystyle O=(h,k)}$ is the circle's center also spelled as centre. The diameter (D) is twice the length of the radius.

## Equations

#### Standard Form

The standard equation for a circle with center ${\displaystyle (h,k)}$ and radius ${\displaystyle r}$ is

${\displaystyle (x-h)^{2}+(y-k)^{2}=r^{2}}$.

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

${\displaystyle x^{2}+y^{2}=r^{2}}$

#### General form

The general form of a circle equation is

${\displaystyle x^{2}+y^{2}+2gx+2fy+c=0}$, 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 ${\displaystyle a}$,

${\displaystyle r=a}$.

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

${\displaystyle r^{2}-2rr_{0}\cos(\theta -\varphi )+r_{0}^{2}=a^{2}}$,
where ${\displaystyle r_{0}}$ is the distance from the circle's center to the origin and ${\displaystyle \varphi }$ 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.

.....

#### Parametric Equations

When the circle's equation is parametrized with respect to ${\displaystyle t}$, the equation becomes

${\displaystyle x=h+r\cos t}$,
${\displaystyle y=k+r\sin t}$.

## Example

Find the center and the radius of the following circle: x2+y2+8x-10y+20=0 find by:

x2+y2+8x-10y+20=0
x2+y2+8x-10y= - 20
(x2+8x)+(y2-10y)= - 20
+16 +25 +16+25
(x2+8x+16)+(y2-10y+25)=21
(x+4)2+(y-5)2=21

Thus:
C(-4,5) radius=${\displaystyle radical(21)}$