Tutorials with detailed solutions to examples and matched exercises on finding equation of a circle, radius and center. Detailed
explanations are also provided.
For more tutorials on equation of circle go here .
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 C(h,k) and
radius r is as follows:
(x  h)^{2} + (y  k)^{2} =
r^{2}
Example 1: Find the
equation of a circle whose center is at (2,  4) and radius 5.
Solution to Example 1:
given
(h , k ) = (2 , 
4)
and r = 5
substitute h, k and r in the standard
equation
(x  2)^{2} + (y  ( 4))^{2} = 5^{2}
(x  2)^{2} + (y + 4)^{2} = 25
Go here and set h, k and r parameters into applet and plot
the circle. Verify graphically that the equation is that of a 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.
Answers.
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 of the
circle is the midpoint of the line segment making the diameter AB.
The midpoint formula is used
to find the coordinates of the center C of the circle.
x coordinate of C = (1 + 3) /2 = 1
y coordinate of C = (2 + 2) / 2 = 2
The radius is half the
distance between A and B.
r = (1/2) ([3  (1)]^{2} + [2  2]^{2} )^{1/2}
= (1/2)(4^{2} + 0^{2})^{1/2}
= 2
The coordinate of C and the radius are
used in the standard equation of the circle to obtain the equation:
(x  1)^{2} + (y  2)^{2 } = 2^{2}
(x  1)^{2} + (y  2)^{2 } = 4
Go here and set h, k and r parameters into 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).
Answers.
Example 3: Find the
center and radius of the circle with equation
x^{2}  4x + y^{2}  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 x^{2}
together and all terms with y and y^{2} together using brackets.
(x^{2}  4x) +( y^{2}  6y) + 9 = 0
We now complete the square
within each bracket..
(x^{2}  4x + 4)  4
+ ( y^{2}  6y + 9)  9
+ 9 = 0
(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} =
2^{2}
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 + y^{2}  8y + 1 = 0
Answers.
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.
center C at (2 , 3)
radius r = (9)^{1/2} = 3
distance from C to P = ([3  (2)]^{2} + [4  3]^{2})^{1/2}
= (5^{2} +1^{2})^{1/2}
= (26)^{1/2}
Since the distance from C to
P is (26)^{1/2 }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
Answers.
More links and references related to the above topics.
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.
