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.

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 $$r^2 - 2 r r_0 \cos(\theta - \varphi) + r_0^2 = a^2$$, 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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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=$$ radical(21)$$