Quote of the Day:
"The greater our knowledge increases, the greater our ignorance unfolds." 
          -- John F. Kennedy 

Objectives:
The student will review the definition of a circle.

The student will apply the general formula for a circle to specific problems.

Given the center and radius of a circle, the student will write the equation for the circle.

1. How do we define a circle (the locus definition)?

   Definition: A circle is the set of points in a plane at 
               a fixed distance from a given point.

   To develop the equation for a circle:
     Call the given point (h, k) – this is the center.
     Let the fixed distance be r – this is the radius.
     Let the set of points be (x, y) – this is the circle.  

     Using either the distance formula or the Pythagorean 
     Theorem, we obtain the general formula for a circle 
     whose center is (h, k) and has a radius of r: 

2. Examples:

3. Circles through points

   How many points are needed to determine a unique circle?

   How many circles may be drawn through one point?

   How many circles may be drawn through two points?

   How many circles may be drawn through three points?

   There is only one circle that may be drawn through three 
   non-collinear points.  

   Given three non-collinear points, how would you find the 
   equation of the circle that passes through them?

   There are at least three different approaches:

   (1) Find the intersection of the perpendicular bisectors 
       of the segments joining the three pairs of points.
       This gives the center.  Then use the distance 
       formula to determine the radius.

   (2) Use the formula

       
       Substitute the coordinates of the three points for
       x and y, then solve the three equations for the 
       three variables h, k, and r.   

   (3) Use the general equation of the circle

       
       Substitute the coordinates of the three points for
       x and y, then solve the three equations for the 
       three variables D, E, and F. 

4. Example
    Determine the equation of the circle that passes 
    through the points (1, 0), (0, 1), and (2, 2).

5. Assignment:
   p. A37 (23c, 26, 30, 33, 36, 40, 45a, 46b, 49)
   Determine the equation of a circle which passes
      through (2,3), (3,2), and (-4,3)

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