Equation of a Circle
Standard form equation of a Circle
Definition 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)
Equation of Circle Interactive HTML5 Applet
Applet on its own page
Explore and discover the standard form equation of a circle using the interactive circle below. To move the circle just click and drag on either of the two points, and the circle's standard form equation will adjust accordingly. You can also see the general form of the circle's equation by clicking 'show general form'
Practice Problems

What is the equation of the circle pictured on the graph below?
Since the radius of this this circle is 1, and its center is the origin, this picture's equation is
$$
(y0)^2 + (x0)^2 = 1^2
\\
y^2 + x^2 = 1
$$

Look at the graph below, can you express the equation of the circle in standard form?

Since the radius of this this circle is 1, and its center is (1,0) , this circle's equation is
$$
(y0)^2 +(x1)^2 = 1^2
\\
y^2 + (x1)^2 = 1
$$

Look at the graph below, can you express the equation of the circle in standard form?
Since the radius of this this circle is 2, and its center is (3,1) , this circle's equation is
$$
(x3)^2 +(y1)^2 = 2^2
\\
(x3)^2 +(y1)^2 = 4
$$
What is the radius of the circles below?
 Y^{2}+X^{2} =9
 Y^{2}+X^{2} =16
 Y^{2}+X^{2} =25
 Y^{2 }+ X^{2} = 11
 Y^{2 }+ X^{2} = a

 $$ \sqrt{9} =3$$
 $$ \sqrt{16} =4$$
 $$ \sqrt{ 25 } =5$$
 $$ \sqrt{11}$$
 $$ \sqrt{a}$$

Look at each standard form equatio below and identify the center and radius.
 (y3)^{2}+(x1)^{2} =9
 (y5)^{2}+(x14)^{2} =16
 (y1)^{2}+(x5)^{2} =25
 (x+2)^{2+}+(y12)^{2} =36
 (y+7)^{2}+(x +5)^{2} =49
 (x +8)^{2}+(y+17)^{2} =49

r = radius
 (1, 3) r = 3
 (14, 5) r = 4
 (5, 1) r = 5
 (2, 12) r = 6
 (5, 7) r = 7
 (8, 17) r = 7

