The standard form equation of a circle is a way to express the definition of a circle on the coordinate plane.
On the coordinate plane, the formula becomes $$(x -h)^2 + (y - k)^2 =r^2 $$
- h and k are the x and y coordinates of the center of the circle
- $$(x-9)^2 + (y-6)^2 =100 $$ is a circle centered at (9,6) with a radius of 10
Examples
General Formula Circle with a center of (4,3) and a radius of 5Another Example
Circle with a center of (2, -1) and a radius of 4Definition of Circle
Definition : A circle is the set of all points that are the same distance, r, from a fixed point.
General Formula: X ^{2} + Y ^{2}=r^{2 }where r is the radius
- Unlike parabolas, circles ALWAYS have X ^{2} and Y ^{2} terms.
X^{2} + Y^{2}=4 is a circle with a radius of 2 (since 4 =2^{2}) - Remember that a circle is a locus of points. A circle is all of the points that are a fixed distance, known as the radius, from a given point, known as the center of the circle.
- Explore the standard equation of a circle using the applet below (Go here for a larger version)
Practice Problems
Since the radius of this this circle is 1, and its center is the origin, this picture's equation is
$$ (y-0)^2 + (x-0)^2 = 1^2 \\ y^2 + x^2 = 1 $$