1. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra Section 3.6: Introduction to Circles

2. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Objectives o Standard form of a circle. o Graphing circles.

3. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Standard Form of a Circle o Two pieces of information are all we need to completely characterize a particular circle: the circle’s center and the circle’s radius. o Suppose is the ordered pair corresponding to the circle’s center, and suppose the radius is given by the positive real number. o Our goal is to develop an equation in the two variables and so that every solution of the equation corresponds to a point on the circle.

4. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Standard Form of a Circle o The main tool that we need to achieve this goal is the distance formula derived in Section 3.1. Since every point on the circle is a distance from the circle’s center, that formula tells us that: o This equation is often presented in the radical free form:

5. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Standard Form of a Circle The standard form of the equation for a circle of radius and center is

6. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 1: Standard Form of a Circle Find the standard form of the equation for the circle with radius and center. Given

7. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 2: Standard Form of a Circle Find the standard form of the equation for the circle with a diameter whose endpoints are and. Step 1: Use the midpoint formula to determine circle’s center. Step 2: Use a slight variation of the distance formula to determine.

8. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 3: Standard Form of a Circle Find the standard form of the equation for the circle that is tangent to the line and whose center is. The word tangent in this context means that the circle just touches the line. It must touch the vertical line at the point. The distance between these two points must then be the radius,. So the equation for this circle is:

9. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Graphing Circles o Given an equation for a circle, we will need to determine the circle’s center and radius and, possibly, graph the circle. o If the equation is given in standard form, this is very easily accomplished. o However, we may have to resort to a small amount of algebraic manipulation in order to determine that a given equation describes a circle and to determine the specifics of that circle. o This is usually done by completing the square.

10. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 4: Graphing Circles Sketch the graph of the circle defined by:

11. HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. Example 5: Completing the Square Sketch the graph of the equation Note: We used the method of completing the square to get the equation in standard form.