General Equation of a Circle
University of Minnesota
General Equation of a Circle
Preliminaries and Objectives
Preliminaries
• Pythagorean Theorem
• Transformation of graphs (shifting horizontally and
vertically)
Objectives
• Find the equation of a circle, given the center and the
radius.
University of Minnesota
General Equation of a Circle
Geometric Definition
A circle is the set of all points located a fixed distance from
some fixed point.
The fixed distance is called the radius of the circle.
The fixed point is called the center of the circle.
University of Minnesota
General Equation of a Circle
Circle centered at the origin
y
(x, y )
5
x
University of Minnesota
General Equation of a Circle
Circle centered at the origin
y
(x, y )
5
y
x
University of Minnesota
x
General Equation of a Circle
Circle centered at the origin
y
(x, y )
5
y
x
x
x 2 + y 2 = 25
University of Minnesota
General Equation of a Circle
Circle centered at the origin
y
(x, y )
r
y
x
x
x2 + y2 = r2
University of Minnesota
General Equation of a Circle
Center at (h, k )
y
(x, y )
3
(3, 2)
x
University of Minnesota
General Equation of a Circle
Center at (h, k )
y
(x, y )
3
(3, 2)
y −2
x −3
x
University of Minnesota
General Equation of a Circle
Center at (h, k )
y
(x, y )
3
(3, 2)
y −2
x −3
x
(x − 3)2 + (y − 2)2 = 9
University of Minnesota
General Equation of a Circle
Center at (h, k )
y
(x, y )
r
(h, k )
y −k
x −h
x
(x − h)2 + (y − k )2 = r 2
University of Minnesota
General Equation of a Circle
Recap
General Form of a Circle:
The circle with center at (h, k ) and radius r has the equation
(x − h)2 + (y − k )2 = r 2
University of Minnesota
General Equation of a Circle
Sample Problem 1
Find the equation of a circle with center at (−2, 1) and radius 4.
y
(x, y )
4
(−2, 1)
x
University of Minnesota
General Equation of a Circle
Sample Problem 1
Find the equation of a circle with center at (−2, 1) and radius 4.
y
(x, y )
4
(−2, 1)
x
(x + 2)2 + (y − 1)2 = 16
University of Minnesota
General Equation of a Circle
Sample Problem 2
Find the center and radius of a circle given by the equation
(x + 6)2 + (y + 3)2 = 4
University of Minnesota
General Equation of a Circle
Sample Problem 2
Find the center and radius of a circle given by the equation
(x + 6)2 + (y + 3)2 = 4
Solution:
Center = (−6, −3); Radius = 2
University of Minnesota
General Equation of a Circle
Credits
Written by:
Mike Weimerskirch
Narration:
Mike Weimerskirch
Graphic Design: Mike Weimerskirch
University of Minnesota
General Equation of a Circle
Copyright Info
c The Regents of the University of Minnesota & Mike
Weimerskirch
For a license please contact http://z.umn.edu/otc
University of Minnesota
General Equation of a Circle