This document provides information about circles, including their geometric and algebraic definitions, how to write the equation of a circle in standard form, and how to graph circles. It discusses that a circle is defined geometrically as the set of all points equidistant from a fixed point, and algebraically as the set of all points with a constant distance from a fixed point. The standard form of a circle equation is given as (x-h)2 + (y-k)2 = r2, where (h,k) is the center and r is the radius. Examples are provided of writing and graphing various circle equations in this standard form.

1. Colleen Beaudoin
February, 2009

2.  Review: The geometric definition relies on a
cone and a plane intersecting it
 Algebraic definition: a set of points in the
plane that are equidistant from a fixed point on
the plane (the center).

3. Find the distance from the center of the circle (h,k)
to any point on the circle (represented by (x,y)).
This is the radius of the circle.
Review the distance formula:
Substitute in the values.
Square both sides to get
the general form of a
circle in center-radius form.
y
x
(h,k)
(x,y)
2 2
2 1 2 1
( ) ( )
d x x y y
   
2 2
( ) ( )
r x h y k
    r
2 2 2
( ) ( )
r x h y k
   

4. Radius (r)
Center (h,k)

5.  Both variables are squared.
 Equation of a circle in center-radius form:
 What makes the circle different from the a line?
 What makes the circle different from the
parabola?

6. center is (3, 1) 4
r
 
center is ( 5,2) 15
r
 
center is (0,2) 3
r 

7. 4. Write the equation of a circle centered at (2,-7)
and having a radius of 5.
(x - 2)2 + (y + 7)2 = 25
5. Describe (x - 2)2 + (y + 1)2 = 0
A point at (2,-1)
6. Describe (x + 1)2 + (y - 3)2 = -1
No graph

8. 7. Write the equation of a circle whose diameter is the
line segment joining A(-3,-4) and B(4,3).
What must you find first?
The center and the radius.
How can you find the center?
The center is the midpoint of the segment.
(½ , - ½ )
How can you find the radius?
The radius is the distance from the center to a
point on the circle. Use the distance formula.
The equation is:
7 2
r =
2
2 2
1 1 49
2 2 2
x y
   
   
   
   

9. 8. Write in center-radius form and sketch:
Hint: You must complete the square.
2 2
6 ___ 4 ___ 12
x x y y
     
2 2
6 4 12
x y x y
   
2 2
6 9 4 4 12 9 4
x x y y
       
2 2
( 3) ( 2) 25
x y
   

10. 1) What’s the standard form of a line?
2) What are the steps for graphing a circle?
3) How can you tell if the graph of an equation
will be a line, a parabola, or a circle?