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Equations of Circles 10.6
CA State Standards
17: Prove theorems using
coordinate geometry.
Lesson Goals
Write the equation of a
circle.
Graph a circle on the
coordinate plane.
ESLRs: Becoming Effective Communicators,
Competent Learners and Complex Thinkers
Previously on Geometry
point-slope form of a linear equation: y - y1 = m( x - x1 )
slope-intercept form of a linear equation: y = mx + b
c 2  a 2  b2 is used to find lengths of a right triangle.
definition
Equation of a Circle
On the coordinate plane a circle is defined by
(x – h)2 + (y – k)2 = r2
(h, k) is the coordinate of the center
r is the radius of the circle
(x, y)
r
(h, k)
Give the center and radius of the circle
a)
 x  4   y  2
2
center  4, 2 
2
 25
c)
radius  5
b)
x   y  1  17
2
2
center  0, 1
radius  17
center  2, 2 
radius  4
example
Write the equation of a circle.
Center: (5, 1)
Radius: 4
 x  h   y  k   r
2
2
2
 x  5   y  1  4
2
2
 x  5   y  1  16
2
2
2
example
Write the equation of a circle.
Center: (4, 3)
Point on Circle: (2, 1)
 x  h   y  k   r
2
2
2
 x  4   y   3   r
2
2
2
 2  4   1  3  r
2
2
2
 2    420
 r
2
2
 x  4    y  3  20
2
2
2
example
 x  2    y  3
 2, 3
Graph the circle:
Center =
Radius =
3
2
2
9
example
 x  3   y  1
3, 1
Graph the circle:
Center =
Radius =
5
2
2
 25
Example: solution graphically2
The equation of a circle is  x  13   y  6   9
2
Are the following points on,
inside or outside the circle?
A 10, 6  on
C
B
A
B 14, 8  inside
C  20, 9  outside
8
solution algebraically (write in column)
If you plug in a point (x, y) and
 x  h   y  k 
2
2
r
 x  h   y  k 
2
 x  h   y  k 
2
2
2
2
then (x, y) is on the circle
r
2
then (x, y) is outside the circle
r
2
then (x, y) is inside the circle
Example: solution algebraically
The equation of a circle is  x  13   y  6   9
2
Are the following points on,
inside or outside the circle?
A 10, 6  B 14, 8 
inside
 3
_
2
2
2
9
9
59
? 9
0? 9
on
14  13  8  6  ?
2
2
1   2  ?
B
2
10  13   6  6 
2
A
C  20, 9 
2
99
 20  13   9  6  ?
2
2
C  7    3 ?
2
outside
2
9
9
58  9
Summary
Explain the equation of a circle.
How can you tell algebraically if a point is
inside, on, or outside a circle?
Today’s Assignment
 p. 638: 8 – 40 e
13
Give the center and radius of the circle.
Give the coordinates of the center, the radius,
and the equation of the circle.
Write the standard equation of the circle
with the given center and radius.
Use the given information to write the
standard equation of the circle.
Graph the equation.
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