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.

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 = r2


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 = 52

                                        (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)(42 + 02)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 =  22

                                             (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

                                            x2 - 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 brackets.

                                            (x2 - 4x) +( y2 - 6y) + 9 = 0

We now complete the square within each bracket..

                                             (x2 - 4x + 4) - 4 + ( y2 - 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 = 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

                                            x2 - 2x + y2 - 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                                         

                                          = (52 +12)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.


  • Interactive HTML5 Math Web Apps for Mobile LearningNew !
    Free Online Graph Plotter for All Devices
    Home Page -- HTML5 Math Applets for Mobile Learning -- Math Formulas for Mobile Learning -- Algebra Questions -- Math Worksheets -- Free Compass Math tests Practice
    Free Practice for SAT, ACT Math tests -- GRE practice -- GMAT practice Precalculus Tutorials -- Precalculus Questions and Problems -- Precalculus Applets -- Equations, Systems and Inequalities -- Online Calculators -- Graphing -- Trigonometry -- Trigonometry Worsheets -- Geometry Tutorials -- Geometry Calculators -- Geometry Worksheets -- Calculus Tutorials -- Calculus Questions -- Calculus Worksheets -- Applied Math -- Antennas -- Math Software -- Elementary Statistics High School Math -- Middle School Math -- Primary Math
    Math Videos From Analyzemath
    Author - e-mail


    Updated: 2 April 2013

    Copyright © 2003 - 2014 - All rights reserved