Chapter 10.1  The Circle

By: Kevin A.

ICM

 

    Johnny went to the grocery store on a Thursday afternoon, and was blown away at the event taking place.  The employees, as well as a few customers, were involved in an intense competition to bake the perfect circular sweet potato pie.  Johnny decides to join in the competition and quickly bakes a nearly perfect circular pie.  Now, the grocery store had a life size picture of a graph in order to determine how perfect each circle was.  When the manager of the grocery store put Johnny's pie up to the graph, it passed through coordinates (1,3), (5,5), and (5,3).  Find the equation of his pie and then identify its center and radius.

 

1. In order to find the equation of the circle/pie, plug the coordinates into the general form of the equation of a circle.  This is: x2 +y2 +Dx+Ey+F=0.  

2.  So, plug (1,3) into this, 1 being x and 3 being y.  This becomes 12 +32 +D +3E +F=0. Then simplify the equation to D+3E+F= -10.

3.  Do the same for (5,5) and (5,3) and the two equations then become respectively, 5D+5E+F= -50; and 5D+3E+F= -34.  

4.  Next, plug the numbers into two matrices in order to find the equation of the circle:  

Use your calculator to find A-1B.

1 3 1
5 5 1
5 3 1 
5.  Put this matrix into the calculator and then, take the 
inverse of it and multiply it by the answer matrix found from
the answers to the three equations found earlier.  This matrix is:
-10
-50
-34 
6.  After taking the inverse of the first matrix and multiplying
it by the second matrix, the answer is:
-6
-8
20

7.  Next, take the answer matrix and plug it into the general form of the equation of a circle. This becomes x2+y2 - 6x - 8y +20=0.

  8.  To find the center and radius, first group together the x terms and the y terms and complete the squares.  To do this, put together x2 - 6x  and y2 - 8y.  Then, take half of the 6x term and then square it.  Do the same for the 8y term.  This brings the equation to x2 - 6x+9+y2 - 8y+16= -20+16+9.  Notice that on the other side of the equal sign, we moved 20 over to become -20 and then added 16 and 9, (the squared terms).

9.  Group them together to get (x-3)2 +(y-4)2 =5.  We added the -20, 16, and 9 to get the 5.  The center is then (3,4) and the radius is the square root of 5.  Always take the opposite sign of the terms in parenthesis for the center and take the square root of the number that the equation is equal to for the radius.  

10.  If you want to graph a circle, first plot the center at (3,4) on the graph and then move up, down, and side to side square root of 5 units.  then connect the dots in a circle. 

- JOHNNY ACTUALLY LOST THE CONTEST BECAUSE HIS RADIUS WAS A SQUARE ROOT, WHEREAS THE LOCAL BANKER HAD A PERFECT CENTER AS WELL AS A RADIUS OF 3, A PERFECT WHOLE NUMBER.     

 

Character Education Quote: "Courage"

"Courage is resistance to fear, mastery of fear - not absence of fear."

-Mark Twain

- This quote means that courage has been perceived as a trait in which fear does not exist at all, but in fact courage is the mastering, and overcoming of fear, rather than not having fear at all. 

Related Web Page: 

http://www.teenoutreach.com/Online_Help/tutorials/math/12/conics/circle.htm

 

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