Equation of Circles Topic Index | Algebra2/Trig Index | Regents Exam Prep Center

Definition:  A circle is a locus (set) of points in a plane equidistant from a fixed point.

 Circle whose center is at the origin Circle whose center is at (h,k) (This will be referred to as the "center-radius form". It may also be referred to as "standard form".) Equation:   Example:  Circle with center (0,0), radius 4                 Graph: Equation:   Example:  Circle with center (2,-5), radius 3                  Graph:

Now, if we "multiply out" the above example we will get:

 When multiplied out, we obtain the "general form" of the equation of a circle.  Notice that in this form we can clearly see that the equation of a circle has both x2 and y2 terms and these terms have the same coefficient (usually 1).

When the equation of a circle appears in "general form", it is often beneficial to convert the equation to "center-radius" form to easily read the center coordinates and the radius for graphing.

Examples:

This conversion requires use of the technique of completing the square.

 We will be creating two perfect square trinomials within the equation. • Start by grouping the x related terms together and the y related terms together.  Move any numerical constants (plain numbers) to the other side. • Get ready to insert the needed values for creating the perfect square trinomials.  Remember to balance both sides of the equation. • Find each missing value by taking half of the "middle term" and squaring.  This value will always be positive as a result of the squaring process. • Rewrite in factored form. You can now read that the center of the circle is at (2, 3) and the radius is .

 2.  How do the coordinates of the center of a circle relate to C and D when the equation of the circle is in the general form                                       ?

Let's make some observations.  Re-examine our previous equations in general form and center-radius form.  Do you see a relationship between the center coordinates and C and D?

 General form Center-radius form C = -4,  D = 10 Center (2, -5) C = -4,  D = -6 Center (2, 3)

It appears that the values of C and D are (-2) times the coordinates of the center respectively.  Why is this occurring?
When is expanded, becomes , where the center term's coefficient doubles the value of
-2.  Remember that while the equation deals with , the actual x-coordinate of the center of this circle is +2.

 3.  Write the equation of a circle whose diameter has endpoints (4, -1) and (-6, 7).
 Find the center by using the midpoint formula.    Find the radius by using the distance formula. Points (-6,7) and (-1,3) were used here.  (d = distance, or radius) Equation:

 4.  Write the equation for the circle shown below if it is shifted 3 units to the right and 4 units up.

A shift of 3 units to the right and 4 units up places the center at the point (3, 4).  The radius of the circle can be seen from the graph to be 5.

Equation:

 Whoa!!!  This equation looks different.  Are we sure this is a circle???In this equation, both the x and y terms appear in squared form and their coefficients (the numbers in front of them) are the same.  Yes, we have a circle here!  We will, however, have to deal with the coefficients of 2 before we can complete the square.

•  group the terms

•  divide through by 2

•  get ready to create perfect squares

•  take half of the "middle term" and square it

•  factor and write in center-radius form

 Topic Index | Algebra2/Trig Index | Regents Exam Prep Center Created by Donna Roberts