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Copyright © 2011 Hawkes Learning Systems.
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Hawkes Learning Systems:
College Algebra
Section 3.6: Introduction to Circles
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Objectives
o Standard form of a circle.
o Graphing circles.
HAWKES LEARNING SYSTEMS
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Copyright © 2011 Hawkes Learning Systems.
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Standard Form of a Circle
o Two pieces of information are all we need to
completely characterize a particular circle: the circle’s
center and the circle’s radius.
o Suppose  h, k  is the ordered pair corresponding to
the circle’s center, and suppose the radius is given by
the positive real number r.
o Our goal is to develop an equation in the two
variables x and y so that every solution  x, y  of the
equation corresponds to a point on the circle.
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Standard Form of a Circle
o The main tool that we need to achieve this goal is the
distance formula derived in Section 3.1. Since every
point on the circle is a distance r from the circle’s
center, that formula tells us that:
r
 x  h   y  k 
2
2
.
o This equation is often presented in the radical free
form:
r   x  h   y  k  .
2
2
2
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Standard Form of a Circle
The standard form of the equation for a
circle of radius r and center  h, k  is
 x  h   y  k 
2
2
r .
2
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Example 1: Standard Form of a Circle
Find the standard form of the equation for the circle
with radius 4 and center  2, 3 .
h2
Given
k  3
r4
 x  2    y   3  
2
 x  2    y  3
2
2
 42
2
 16
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Example 2: Standard Form of a Circle
Find the standard form of the equation for the circle
with a diameter whose endpoints are  5,3 and 11,9  .
Step 1: Use the midpoint
 5  11 3  9 
formula to
,
 h, k   
  8,6 
2 
 2
determine
circle’s center.
Step 2: Use a slight variation
of the distance
formula to
determine r 2.
r   5  8   3  6 
2
2
2
r 2  9  9  18
18   x  8   y  6
2
2
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Example 3: Standard Form of a Circle
Find the standard form of the equation for the circle
that is tangent to the line x  1 and whose center
is  2,7 .
The word tangent in this context means that the circle
just touches the line x  1. It must touch the vertical
line at the point 1,7  . The distance between these two
points must then be the radius, r  3 . So the equation
for this circle is:
 x  2   y  7 
2
2
9
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Graphing Circles
o Given an equation for a circle, we will need to
determine the circle’s center and radius and,
possibly, graph the circle.
o If the equation is given in standard form, this is very
easily accomplished.
o However, we may have to resort to a small amount of
algebraic manipulation in order to determine that a
given equation describes a circle and to determine
the specifics of that circle.
o This is usually done by completing the square.
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Example 4: Graphing Circles
Sketch the graph of the circle defined by:
 x  2    y  3  4
2
2
2
x

2

y


3

2

   
 h, k    2, 3 and r  2
2
2
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Example 5: Completing the Square
Sketch the graph of the equation
x 2  y 2  8 x  2 y  1
2
2
x

8
x

16

y

   2 y  1  1  16  1
 x  4    y  1  16
 h, k    4,1
2
2
r4
Note: We used the method of
completing the square
to get the equation in
standard form.