Tutorial on Equation of Circle

Tutorials with detailed solutions to examples and matched exercises on finding equation of a circle, radius and center. Detailed explanations are also provided. More tutorials on on equations of circles: (2) and (3) are included.

Definition

A circle is the set of points equidistant from a point C(h,k) called the center . The fixed distance r from the center to any point on the circle is called the radius .
The standard
equation of a circle with center at C(h,k) and radius r is as follows:
(x - h)2 + (y - k)2 = r2


Example 1

Find the equation of a circle whose center is at (2, - 4) and radius 5.

Solution to Example 1


Substitute (h , k ) by (2 , - 4) and r by 5 in the standard equation to obtain
(x - 2)
2 + (y - (- 4))2 = 52Simplify
(x - 2)
2 + (y + 4)2 = 25
Set h, k and r parameters into
this applet and plot the circle. Verify graphically that the equation is that of the circle with the given center and radius.

Matched Exercise 1

Find the equation of a circle whose center is at (2 , - 4) and radius 3.

Solution.

Example 2

Find the equation of a circle that has a diameter with the endpoints given by the points A(-1 , 2) and B(3 , 2).

Solution to Example 2


The center C of the circle is the midpoint of the line segment making the diameter AB. We first use the midpoint formula to find the coordinates of C.
C ( (-1 + 3) / 2 , (2 + 2) / 2) = C(1,2)
The radius r is half the distance between A and B. Hence
r = (1 / 2) √ ( [3 - (-1)]
2 + [2 - 2]2 )
= (1/2)√(4
2 + 02)
= 2
The coordinate of C and the radius r are used in the standard equation of the circle to obtain the equation:
(x - 1)
2 + (y - 2)2 = 22
Simplify
(x - 1)
2 + (y - 2)2 = 4
Set the h, k and r parameters
into this applet and plot the circle. Verify graphically that the equation is that of a circle with the diameter as given above.

Matched Exercise 2

Find the equation of a circle that has a diameter with the endpoints given by A(0 , -2) and B(0 , 2).

Solution

Example 3

Find the center and radius of the circle with equation
x
2 - 4x + y2 - 6y + 9 = 0

Solution to Example 3


In order to find the center and the radius of the circle, we first rewrite the given equation into the standard form as given above in the definition. Put all terms with x and x2 together and all terms with y and y2 together using parentheses.
(x
2 - 4x) +( y2 - 6y) + 9 = 0
We now complete the square within the parentheses.
(x
2 - 4x + 4) - 4 + ( y2 - 6y + 9) - 9 + 9 = 0
Which may be written as.
(x - 2)
2  + ( y - 3)2 - 4 - 9 + 9 = 0
Simplify and write in standard form
(x - 2)
2 + ( y - 3)2 = 4
(x - 2)
2  + ( y - 3)2 = 22
We now compare this equation and the standard equation to obtain.
center at C(h , k) = C(2 , 3)
and radius r = 2

Matched Exercise 3

Find the center and radius of the circle with equation
x
2 - 2x + y2 - 8y  + 1 = 0

Solution

Example 4

Is the point P(3 , 4) inside, outside or on the circle with equation
(x + 2)
2 + ( y - 3)2 = 9

Solution to Example 4


We first find the distance from the center of the circle to point P.
Using the given equation the center C is at (-2 , 3)
and the radius r = √9 = 3
distance from C to P is equal to: √([3 - (-2)]
2 + [4 - 3]2)
= √(5
2 +12)
= √(26)
Since the distance from C to P is √(26) which approximately equal to 5.1 is greater than the radius r = 3, point P is outside the circle. You can check your answer graphically using this applet

Matched Exercise 4

Is the point P(-1 , -3) inside, outside or on the circle with equation
(x - 1)
2  + ( y + 3)2  =  4

Solution

For more tutorials on equation of circle go here .

More References and links related to the equation of a circle.


Find x and y intercepts of Circles - Calculator: A calculator to calculate the x and y intercepts of the graph of a circle given its center and radius.
Find center and the radius of a Circle: Calculates the coordinates of the center and radius of a circle given its equation.
Match Equations of Circles to Graphs. Excellent interactive activity where equations of circles are matched to graphs.
Tutorials on equation of circle.
Interactive tutorial on equation of circle.
Three Points Circle Calculator.