EQUATIONS OF A CIRCLE
Name_____________________________ Date__________ Period ________
PART ONE- Deriving the Equation of a Circle
Consider the circle in a coordinate plane that has its center C (h, k) and that has radius r.
A. Let P be any point on the circle and let the coordinates of P be (x, y). Create a right triangle by drawing
a horizontal line through C and a vertical line through P, as shown.
What are the coordinates of point A? _________
Write expressions for the lengths of the legs of Δ CAP.
CA = ___________ PA = ______________
B. Use the Pythagorean Theorem to write the
relationship among the side lengths of Δ CAP.
_________________ + ________________ =
______________
Write the equation of a circle with center (h, k) and radius r.
Suppose a circle has its center at the origin. What is the
equation of the circle in this case?
___________________________________
PART TWO – Graph a circle from an equation.
1.) Write the equation in standard form
𝑥2 + 𝑦2 = 𝑟2
or
(x-h)2 + (k-h)2 = r2
EXAMPLE: (x – 2)2+ (x – 1)2= 16
2.) Determine the center (h,k), graph center (2,1)
3.) Determine the radius: √16 = 4
4.) Graph four points on the circle
5.) Connect coordinate points.
PART THREE: Write the Equation of a circle from a
graph.
1. Determine the center of the circle (h,k)
2. Determine the radius distance between (h,k) and a
point on the circle (x,y)
d = √(𝑥 − ℎ)2 + (𝑥 − 𝑘)2
3. Write in standard form of equation of circle.
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Graph the circle: (x + 3)2 + (y - 2)2 = 9
Identify 4 points on the circle.
(_____, _____) (_____, _____) (_____, _____) (_____, _____) y
x
Identify the radius and the center of each equation below.
Eq. #
Standard Graphing Equation
1
( x – 2 ) + ( y – 5 ) = 144
2
2
Radius
Center
2
2
2
( x – 0 ) + ( y – 7 ) = 625
2
2
2
2
2
2
3
( x – 15 ) + ( y + 8 ) = 289
4
( x + 11 ) + ( y + 6 ) = 324
5
( x + 4 ) + ( y – 9 ) = 196
Given each center and radius below, write the equation of the circle in standard graphing form.
Eq. #
Center
Radius
1
( -5 , 12 )
13
2
( 9 , -12 )
15
3
( -7 , 0 )
10
4
( 11 , 14 )
9
5
( -6 , -8 )
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Standard Graphing Equation
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PART FOUR – Writing a Coordinate Proof
Prove or disprove the point (1,√15) lies on the circle that is centered at the origin and contains the point (0, 4).
A. Plot a point at the origin and at (0, 4). Use these to help you draw the center centered at the origin
that contains (0,4).
y
B. Determine the radius : r =___________
C. Use the radius and the coordinates of the center to write the
equation of the circle.
D. Substitute the x- and y- coordinates of the point (1,√15) in
the equation of the circle to check whether they satisfy the
equation.
__________2 + _________2 = 16 (substitute)
__________+ _________ = 16 (simplify)
E. So the point (1,√15) lies on the circle because
_______________________________________________________
F. Explain how to determine the radius of the circle.
_________________________________________________________________________________
_____________________________
G. Name another point with noninteger coordinates that lies on the circle. Explain.
_______________________________________________
H. Explain how you can prove that point (2,√5) does not lie on the circle.
_______________________________________________________
Write an equation in standard form of each circle described below:
1. End points of a diameter are (3, 2) and (5, –4).
2.
r  5 , Center (-3, 5)
3.
r  8 , Center (-1, -6)
4.
r  12 , Center (4, -2)
5.
r  12 , Center ( 12 , 0)
3
x
Name_____________________________________ Date_________ Period______
Note: If r2 is not a perfect square then leave r in simplified radical form but use the decimal equivalent for
graphing. Example: 12  2 3  3.46
1) Graph the following circle:
a. (x - 3)2 + (y + 1)2 = 4
b. (x – 2)2 + (y – 5)2 = 9
c. (y + 4)2 + (x + 2)2 = 16
2) For each circle: Identify its center and radius.
a. (x + 3)2 + (y – 1)2 = 4
b. x2 + (y – 3)2 = 18
Center: ____________
Radius:_____________
c. (y + 8)2 + (x + 2)2 = 72
Center: ___________
Radius:___________
Center:_____________
Radius:_____________
3) Write the equation of the following circles:
4) Compare and contrast the following pairs of circles
a. Circle #1: (x - 3)2+ (y +1)2 = 25
b. Circle #1: (y + 4)2+ (x + 7)2 = 6
Circle #2: (x + 1)2 + (y - 2)2 = 25
Circle #2: (x + 7)2 + (y + 4)2 = 36
8) Give the equation of the circle whose center is
(5,-3) and goes through (2,5)
9) Give the equation whose endpoints of a diameter
at (-4,1) and (4, -5)
10) Give the equation of the circle whose center is
(4,-3) and goes through (1,5)
11) Give the equation whose endpoints of a
diameter at (-3,2) and (1, -5)
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Math II
Equations of a Circle
Name_______________________________________Date_________Period_____
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