The Circle

  1. Conic Sections

    A conic section is formed by intersecting a plane with a cone.  The different possible conic sections are the circle, parabola, ellipse, and the hyperbola.


  2. Circles

    A circle is the set of points in a plane a fixed distance from a point.  By the Pythagorean Theorem, we have that the distance r from the center (h,k) of the circle to a point (x,y) on the circle is

            r = [(x - h)2 + (y - k)2]1/2

    or


    (x - h)2 + (y - k)2 = r2



    Example:      

    Find the equation of the circle with center (2,1) and  radius 4.

    Solution:   

    We have:  

            (x - 2)2 + (y - 1)2 = 42 = 16

    Exercise:


    1. Find the equation of the circle with center (1,3) and passing through the point (7,11)

      Graph the following:

    2. (x - 2)2 + (y + 1)2 = 9

    3.  x2 - 2x + y2 + 6y = 14

    4.  x2 + y2 + 4x - 4y = 9

    5.  x2 + y2 + 6x + 2y = 29

    6. Find the area between the circles:

              x2 + y2 - 6x + 4y = 12

      and

              x2 + y2 - 6x + 4y = 23

     

  3. Example:  Circles and Tangent Lines

    Find the equation of the circle that has center (3,-2) and is tangent to the line 

            x + 2y = 4


    Solution

    Since the line segment joining the center of the circle and the point where the line meets the circle is perpendicular to the line 

            x + 2y = 4

    this segment has slope equal to the negative reciprocal of the slope of 

            x + 2y = 4 

    or 

            y = -1/2 x + 2

    Hence this segment has slope equal to 2.  The segment lies on the line

            y + 2 = 2(x - 3)
    or

            y = 2x - 8

    The point of tangency is given by the intersection of the tangent line with this segment:

            -1/2 x + 2 = 2x - 8

     so

            10 = 2x + 1/2 x

     or

            20 = 4x + x = 5x

     hence

            x = 4     and     y = 2(4)- 8 = 0.
    Now use the distance formula to find the radius of the circle:

            r = [(0 - -2)2 + (4 - 3)2]1/2 =

    The equation of the circle is

            (x - 3)2 + (y + 2)2 = 5

 

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