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. Template:TextBox

Equations

Standard Form

The standard 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

$r^2-2r{r}_0\cos (\theta -\phi )+r_0^2=a^2$,
where $r_0$ is the distance from the circle's center to the origin and $\phi$ 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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Parametric Equations

When the circle's equation is parametrized with respect to $t$, the equation becomes

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

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=$radical(21)$