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.

Equations

General Form

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

(x − h)² + (y − k)² = r².

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

x² + y² = r²

General form

The general form of a circle equation is

x² + y² + 2gx + 2fy + c = 0, where

<-g,-f>

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 r₀ is the distance from the circle's center to the origin and φ 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 t, the equation becomes

x = h + rcos t,
y = k + rsin 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)