Conic Sections/Circle

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Contents 1 Definition 2 Equations 2.1 General Form 2.2 Standard form 2.3 Polar Coordinates 2.4 Parametric Equations 3 Example

[edit] 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.

[edit] Equations

[edit] General Form

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

(x − h)² + (y − k)² = r².

The radius must be greater than 0. If the radius is zero, the graph is a single point. This is a degenerate case.

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

[edit] Standard form

The standard form of a circle equation is

Ax² + Cy² + Dx + Ey + F = 0, where A = C.

[edit] 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

,
where r₀ is the distance from the circle's center to the origin and 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.

.....

[edit] Parametric Equations

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

x = h + rcost,
y = k + rsint.

[edit] Example

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

x²+y²+8x-10y+20=0
-20-20
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)