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 5 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
$$ (y-0)^2 + (x-0)^2 = 1^2 \\ y^2 + x^2 = 1 $$
Since the radius of this this circle is 1, and its center is (1,0) , this circle's equation is
$$ (y-0)^2 +(x-1)^2 = 1^2 \\ y^2 + (x-1)^2 = 1 $$
Since the radius of this this circle is 2, and its center is (3,1) , this circle's equation is
$$ (x-3)^2 +(y-1)^2 = 2^2 \\ (x-3)^2 +(y-1)^2 = 4 $$
Y^{2}+X^{2} = 9
$$ \sqrt{9} =3$$
Y^{2}+X^{2} = 16
$$ \sqrt{16} = 4$$
Y^{2}+X^{2} = 25
$$ \sqrt{ 25 } = 5 $$
Y^{2 }+ X^{2} = 11
$$ \sqrt{11}$$
Y^{2 }+ X^{2} = a
$$ \sqrt{a}$$
(y-3)^{2}+(x-1)^{2} = 9
(1, 3) r = 3
(y-5)^{2}+(x-14)^{2} = 16
(14, 5) r = 4
(y-1)^{2}+(x-5)^{2} = 25
(5, 1) r = 5
(x+2)^{2+}+(y-12)^{2} = 36
(-2, 12) r = 6
(y+7)^{2}+(x +5)^{2} = 49
(-5, -7) r = 7
(x +8)^{2}+(y+17)^{2} = 49
(-8, -17) r = 7