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Hawkes Learning Systems: College Algebra

Hawkes Learning Systems: College Algebra. Section 3.6: Introduction to Circles. Objectives. Standard form of a circle. Graphing circles. . Standard Form of a Circle.

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Hawkes Learning Systems: College Algebra

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  1. Hawkes Learning Systems:College Algebra Section 3.6: Introduction to Circles

  2. Objectives • Standard form of a circle. • Graphing circles.

  3. Standard Form of a Circle • Two pieces of information are all we need to completely characterize a particular circle: the circle’s center and the circle’s radius. • Suppose is the ordered pair corresponding to the circle’s center, and suppose the radius is given by the positive real number . • 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. Standard Form of a Circle • 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: • This equation is often presented in the radical free form:

  5. Standard Form of a Circle The standard form of the equation for a circle of radius and center is

  6. Example 1: Standard Form of a Circle Find the standard form of the equation for the circle with radius and center . Given

  7. 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. 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. Graphing Circles • Given an equation for a circle, we will need to determine the circle’s center and radius and, possibly, graph the circle. • If the equation is given in standard form, this is very easily accomplished. • 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. • This is usually done by completing the square.

  10. Example 4: Graphing Circles Sketch the graph of the circle defined by:

  11. 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.

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