Section 6-2:  Equations of Circles
 
 
Definition of a Circle
 
A circle is the set of all points in a plane equidistant from a fixed point called the center point.  We can derive the equation directly from the distance formula.  If we place the center point on the origin point, the equation of a circle with center point (0, 0) and radius r is:
x2 + y2 = r2
Look at the family of circles drawn:
 
A simple translation of the circle equation becomes:
(x - h)2 + (y - k)2 = r2
With center at (h, k) and radius r.
Here are some examples:

Sample Problems
 
1)  Find the center, radius and graph the equation:  (x - 2)2 + (y + 5)2 = 17
 
             Solution:  Center point :  (2, -5), radius =   The graph is below:
 
2)  x2 + y2 - 8x + 4y - 8 = 0  Find center, radius and graph.
 
             Solution:  We need to put the equation into the correct form.  We will do this by completing the square!!
                 (x2 - 8x          ) + (y2 + 4y     ) = 8  Complete the square! See 1.6
                 (x2 - 8x + 16) + (y2 + 4y + 4) = 8 + 16 + 4
                 (x - 4)2 + (y + 2)2 = 28  Now it's in the correct form!!
                 Center point (4, -2) with radius =   The graph is:

 
3)  Find the intersection of the line y = x - 1 and the circle x2 + y2 = 25.
 
             Solution:  A line could intersect a circle twice or once (if it is tangent) or not intersect at all.  To find the intersection use substitution.  Replace y in the circle equation with x - 1.
 
                 x2 + (x - 1)2 = 25
                 x2 + x2 - 2x + 1 = 25
                 2x2 - 2x - 24 = 0
                 x2 - x - 12 = 0  Now factor
                 (x - 4)(x + 3) = 0
                 x = 4 or x = -3
                 Substituting back in to find y gives the following points:
                 (4, 3) and (-3, -4)  If you check these points in both equations, you will discover they make both true.  The graph of the system is:
What would it mean if the solutions were imaginary?

4)  Sketch the graph of 
 
                 Solution:  The above graph is part of a circle.  Why?  Because it only includes the positive values for y.  Thus it is the top half of a circle with radius 4.  It is a semi-circle.  By only studying this part of the circle, it makes it a function.  It now passes the vertical line test.  Circles are not functions!  Here is the graph:

 
That's it for circles.  Let's head toward Ellipses!