| Circles |
-- Section 2.4 -- |
| The Standard Equation of a Circle*: | (x - h)2 + (y - k)2 = r2 |
| General Form of the Equation of a Circle*: | x2 + y2 + cx + dy + e = 0 |
|
*These names may vary from book to book. |
|
|
Find the equation in general form of a circle with the center at (0,0) and radius of 6. |
|
Solve |
Step |
| r = 6 and (h, k) = (0, 0) | (0,0) is the origin |
| (x - h)2 + (y - k)2 = r2 | Use Standard Equation form |
| (x - 0)2 + (y - 0)2 = 62 | Simplify |
| x2 + y2 = 36 | Set equal to zero |
| x2 + y2 - 36 = 0 | General Form |
|
|
|
 |
Find the equation in general form of a circle with the center at (0,0) and radius of 4. |
 |
[Solution] |
|
|
|
||
 |
Find the equation in general form of a circle with the center at (-3, 2) and radius of 6. |
|
Solve |
Step |
| r = 6 and (h, k) = (-3, 2) | Plug in the values |
| (x - h)2 + (y - k)2 = r2 | Use Standard Equation form |
| (x - (-3))2 + (y - 2)2 = 62 | Simplify |
| (x + 3)2 + (y - 2)2 = 36 | Put in general form |
| x2 + 6x + 9 + y2 - 4y + 4 = 36 | Multiply out |
| x2 + y2 + 6x - 4y + 13 = 36 | Combine like terms and reorder terms |
| x2 + y2 + 6x - 4y - 23 = 0 | Set equal to zero for general form |
| General Form: x2 + y2 + cx + dy + e = 0 | |
|
|
|
|
Find the equation in general form of a circle with the center at (4, -1) and radius of 5. |
|
[Solution] |
|
|
|
||
 |
Given: x2
+ y2 + 10x
- 3y + 3 = 0 Put in the Standard Equation form. |
|
Solve |
Step |
|
x2 + y2 + 10x - 3y + 3 = 0 |
Group like terms |
|
x2 + 10x + y2 - 3y = -3 |
Subtract 3 from both sides |
|
x2 + 10x + 52 + y2 - 3y + (3/2)2 = -3 + 52 + (3/2)2 * |
Take half the middle term and square it. Note half of 3 is 3/2 |
|
(x + 5)2 + (x - 3/2)2 = -3 + 25 + 9/4 |
Complete the
squares!! Add those fractions... |
|
(x + 5)2 + (x - 3/2)2 = 97/4 |
|
|
Standard Equation form: |
(x - h)2 + (y - k)2 = r2 |
| Multiply (x + 5)2 + (x - 3/2)2 = 97/4 to verify | |
|
The center is (-5, 3/2) Radius is |
|
| *Click here for completing the square help with this step. | |
|
|
|
 |
Given: x2
+ y2
- 5x
+ 4y -
2 = 0 Put in the Standard Equation form. |
|
[Solution] |
|
|
|
||
 |
Graph the equation: x2 + y2 - 4y - 21 = 0 |
|
Solve |
Step |
|
|
x2 + y2 - 4y - 21 = 0 |
Put in the Standard Equation form | |
|
x2 + y2 - 4y = 21 |
Add 21 to both sides | |
|
x2 + y2 - 4y + 4 = 21 + 4 |
Take half the middle term and square it. | |
|
x2 + (y - 2)2 = 25 |
Standard Equation form | |
| (x - h)2 + (y - k)2 = r2 | ||
|
The center is (0, 2) and r = 5 |
||
|
|
Locate
the center (0,2)
Then count 5 units along the x and y direction. Draw a beautiful circle. |
|
|
|
||
 Graph
the equation: x2 + 6x
+ y2 -
7 = 0 |
||
|
[Solution] |
|
|
|
||
Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada
