CHAPTER 10, SECTION 1
THE CIRCLE
BY: IJE (EJ) O.
ICM HONORS
Objective: At the end of this section, you should be able to:
form an equation using the general form of the equation of a circle
write the equation in standard form
identify the center and the radius in the standard form of the equation
graph a circle.
Things to know:
* The standard form of the equation of a circle with radius r and center at (h,k) is
(x - h)2 + (y - k)2 = r2
* The general form of the equation of a circle is x2 + y2 + Dx + Ey + F = 0; where D, E, & F are constants.
THE PROBLEM:
Once Sandy visited Washington DC and discovered that the White House, the Washington Monument, and the Pentagon are located at the coordinates (-2,1),(6,-3), and (0, 5) respectively. Given this information, find the following:
The equation of the circle that passes through these points.
The standard form of the circle.
The center and the radius.
To find the equation of the circle, use the general form of the circle:
(-2)2 + (1)2 + D(-2) + E(1) + F = 0
---- 4 + 1 - 2D + E + F = 0
---- -2D + E + F = -5
(6)2 + (-3)2 + D(6) + E(-3) + F = 0
---- 36 + 9 + 6D - 3E + F = 0
---- 6D - 3E + F = -45
(0)2 + (5)2 + D(0) + E(5) + F = 0
---- 0 + 25 + 5E + F = 0
---- 5E + F = -25
You now have three equations:
-2D + E + F = -5
6D - 3E + F = -45
5E + F = -25
To find the values of D, E, & F
Plug in the coefficients for the letters only into matrix A, then plug in -5, -45, -25 into matrix B, and find the inverse. Your matrix in A and B should look like this respectively:
| -2 | 1 | 1 |
| 6 | -3 | 1 |
| 0 | 5 | 1 |
| -5 |
| -45 |
| -25 |
You find the inverse by plugging in A-1B into your calculator. You should end up with:
| -6 |
| -2 |
| -15 |
Therefore, -6, -2, -15 are D, E, and F respectively. Now you can form your equation:
x2 + y2 - 6x - 2y -15 = 0
To find the standard form of the equation, get it by completing the square.
(x2 - 6x + 9) + (y2 - 2y + 1) = 15 + 9 + 1
(x - 3)2 + (y - 1)2 = 25
Now that the equation is in the standard form,
The center is (3, 1), and
The radius is the square root of 25 which is 5.
To graph this equation, plot the points (3, 1) on the graph as your center. Since the radius is 5, move up and down 5 units from the center point and left and right 5 units from the center point. Then connect the four dots to form a circle with the center points in the middle.
" Success
is like a ladder,
And no one has ever climbed a ladder
With their hands in their pockets."
--- Zig Ziglar
The purpose of climbing a ladder is to get to the top. And climbing a ladder requires work. Climbing a ladder is similar to success. To get to the top, you have to work. It is your responsibility to succeed. Everyone can succeed. All you need is determination.
Use this web site to learn more on the circles.
http://www.alltel.net/~okrebs/page61.html