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					10-2        Circles

    Warm Up
    Find the slope of the line that connects
    each pair of points.

                             1
    1. (5, 7) and (–1, 6)    6


    2. (3, –4) and (–4, 3)   –1




Holt Algebra 2
10-2        Circles

    Warm Up
    Find the distance between each pair of
    points.


    3. (–2, 12) and (6, –3)   17


    4. (1, 5) and (4, 1)      5




Holt Algebra 2
10-2        Circles

                  Objectives
   Write an equation for a circle.
   Graph a circle, and identify its center
   and radius.




Holt Algebra 2
10-2        Circles

                  Vocabulary
   circle
   tangent




Holt Algebra 2
10-2        Circles

        A circle is the set of points in a plane that are
        a fixed distance, called the radius, from a fixed
        point, called the center. Because all of the
        points on a circle are the same distance from
        the center of the circle, you can use the
        Distance Formula to find the equation of a
        circle.




Holt Algebra 2
10-2        Circles
 Example 1: Using the Distance Formula to Write the
                Equation of a Circle
  Write the equation of a circle with center (–3, 4)
  and radius r = 6.
   Use the Distance Formula with (x2, y2) = (x, y),
   (x1, y1) = (–3, 4), and distance equal to the radius, 6.

                                Use the Distance Formula.

                                Substitute.

                                Square both sides.


Holt Algebra 2
10-2        Circles
                 Check It Out! Example 1

  Write the equation of a circle with center (4, 2)
  and radius r = 7.
  Use the Distance Formula with (x2, y2) = (x, y),
  (x1, y1) = (4, 2), and distance equal to the radius, 7.


                               Use the Distance Formula.

                               Substitute.

                               Square both sides.



Holt Algebra 2
10-2        Circles

  Notice that r2 and the center are visible in the equation
  of a circle. This leads to a general formula for a circle
  with center (h, k) and radius r.




Holt Algebra 2
10-2        Circles


     Helpful Hint
      If the center of the circle is at the origin, the
      equation simplifies to x2 + y2 = r2.




Holt Algebra 2
10-2        Circles
       Example 2A: Writing the Equation of a Circle

   Write the equation of the circle.

   the circle with center (0, 6) and radius r = 1

       (x – h)2 + (y – k)2 = r2      Equation of a circle

       (x – 0)2 + (y – 6)2 = 12 Substitute.

                 x2 + (y – 6)2 = 1




Holt Algebra 2
10-2        Circles
       Example 2B: Writing the Equation of a Circle

   Write the equation of the circle.
   the circle with center (–4, 11) and containing
   the point (5, –1)

                              Use the Distance Formula to
                              find the radius.



 (x + 4)2 + (y – 11)2 = 152   Substitute the values into the
                              equation of a circle.

 (x + 4)2 + (y – 11)2 = 225
Holt Algebra 2
10-2        Circles
                     Check It Out! Example 2

   Find the equation of the circle with center (–3, 5)
   and containing the point (9, 10).

                                   Use the Distance Formula
                                   to find the radius.



   (x + 3)2 + (y – 5)2 = 132       Substitute the values into
                                   the equation of a circle.
                 2      2
   (x + 3) + (y – 5) = 169



Holt Algebra 2
10-2        Circles

     The location of points in relation to a circle can be
     described by inequalities. The points inside the
     circle satisfy the inequality (x – h)2 + (y – k)2 < r2.
     The points outside the circle satisfy the inequality
     (x – h)2 + (y – k)2 > r2.




Holt Algebra 2
10-2        Circles
           Example 3: Consumer Application
   Use the map and information given in Example 3
   on page 730. Which homes are within 4 miles of
   a restaurant located at (–1, 1)?
   The circle has a center (–1, 1) and
   radius 4. The points insides the circle
   will satisfy the inequality (x + 1)2 +
   (y – 1)2 < 42. Points B, C, D and E
   are within a 4-mile radius .
   Check Point F(–2, –3) is near the boundary.
                 2       2     2
   (–2 + 1) + (–3 – 1) < 4
   (–1)2 + (–4)2 < 42
   1 + 16 < 16 x Point F (–2, –3) is not inside the circle.
Holt Algebra 2
10-2        Circles
                     Check It Out! Example 3
   What if…? Which homes are within a 3-mile
   radius of a restaurant located at (2, –1)?
   The circle has a center (2, –1) and
   radius 3. The points inside the circle
                                      2
   will satisfy the inequality (x – 2) +
   (y + 1)2 < 32. Points C and E are
   within a 3-mile radius .
    Check Point B (1, 2) is near the boundary.
                 2        2     2
     (1 – 2) + (2 + 1) < 3
     (–1)2 + (3)2 < 32
     1+9<9 x           Point B (1, 2) is not inside the circle.
Holt Algebra 2
10-2        Circles

        A tangent is a line in the same plane as the
        circle that intersects the circle at exactly one
        point. Recall from geometry that a tangent to a
        circle is perpendicular to the radius at the point
        of tangency.



        Remember!
        To review linear functions, see Lesson 2-4.




Holt Algebra 2
10-2        Circles
      Example 4: Writing the Equation of a Tangent

   Write the equation of the line tangent to the
   circle x2 + y2 = 29 at the point (2, 5).

   Step 1 Identify the center and radius of the circle.

           From the equation x2 + y2 = 29, the circle has
           center of (0, 0) and radius r =  .




Holt Algebra 2
10-2        Circles
                   Example 4 Continued
   Step 2 Find the slope of the radius at the point of
   tangency and the slope of the tangent.

                        Use the slope formula.

                        Substitute (2, 5) for (x2 , y2 )
                        and (0, 0) for (x1 , y1 ).
                                                      5
                        The slope of the radius is    2    .

    Because the slopes of perpendicular lines are
                                                               2
    negative reciprocals, the slope of the tangent is –        5
                                                                   .
Holt Algebra 2
10-2        Circles
                  Example 4 Continued

   Step 3 Find the slope-intercept equation of the
   tangent by using the point (2, 5) and the slope
   m=– 2.5


                        Use the point-slope formula.

                                                         2
                       Substitute (2, 5) (x1 , y1 ) and – 5 for m.

                        Rewrite in slope-intercept form.



Holt Algebra 2
10-2        Circles
                    Example 4 Continued

       The equation of the line that is tangent to
       x2 + y2 = 29 at (2, 5) is              .


       Check Graph the
       circle and the line.




Holt Algebra 2
10-2        Circles
                 Check It Out! Example 4

   Write the equation of the line that is tangent
   to the circle 25 = (x – 1)2 + (y + 2)2, at the
   point (1, –2).

   Step 1 Identify the center and radius of the circle.

    From the equation 25 = (x – 1)2 +(y + 2)2, the
    circle has center of (1, –2) and radius r = 5.




Holt Algebra 2
10-2        Circles
                 Check It Out! Example 4 Continued

   Step 2 Find the slope of the radius at the point of
   tangency and the slope of the tangent.

                           Use the slope formula.

                            Substitute (5, –5) for (x2 , y2 )
                            and (1, –2) for (x1 , y1 ).
                                                         –3
                           The slope of the radius is    4
                                                                .

    Because the slopes of perpendicular lines are
    negative reciprocals, the slope of the tangent is               .
Holt Algebra 2
10-2        Circles
                 Check It Out! Example 4 Continued

   Step 3. Find the slope-intercept equation of the
   tangent by using the point (5, –5) and the slope                      .


                            Use the point-slope formula.

                                                                     4
                            Substitute (5, –5 ) for (x1 , y1 ) and   3
                            for m.
                            Rewrite in slope-intercept form.



Holt Algebra 2
10-2        Circles
                 Check It Out! Example 4 Continued

   The equation of the line that is tangent to 25 =
   (x – 1)2 + (y + 2)2 at (5, –5) is            .


        Check Graph the
        circle and the line.




Holt Algebra 2

				
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