A circle is the set of all points ( , ) in the Cartesian plane that are a fixed distance r from a fixed point ( , ) .
The fixed distance r is called the radius of the circle and the fixed point ( , ) is called the center of the circle.
To derive the equation of a circle, we use the distance formula.
A circle with center ( , ) and radius r k h r
The standard form of an equation of a circle with center ( , ) and radius r is
 x h
  y k
 2  r
2
.
The standard form of an equation of a circle centered at the origin with radius r x
2  y
2  r
2
.
Objective 1 : Writing the Standard Form of an Equation of a Circle
Note that we can use the midpoint formula to determine the center of a circle given the endpoints of a diameter of the circle. The radius can be found using the distance formula.
2.2.7
Find the standard form of the equation of the circle with endpoints of a diameter at the points _____ and _____.
2.2.9
Write the equation in standard form for a circle with center ______ and tangent to the x-axis.

Objective 2 : Sketching the Graph of a Circle
Once we know the center and radius of a circle, we can easily graph the circle. For additional points, find any intercepts and plot the points.
Note that the ycoordinate of the center of the circle
 x

1
  y

2
 2 
9 ( k
 
2 ) is negative because
 y

2
  y
   2
( 2) .
Objective 3 : Converting the General Form of a Circle into Standard Form
The general form of the equation of a circle is
, , , , and E are real numbers and A

B .
Ax
2 
By
2 
Cx

Dy
 
0 where
By completing the square, the equation of a circle can be rewritten from general form to standard form.
2.2.22
Find the center, radius, and intercepts of the circle with the given equation and then sketch the graph.