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 + y- 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:  x2y2 + 10x - 3y  + 3 = 0
Put in
the Standard Equation form.

Solve

                Step

x2y2 + 10x - 3y  + 3 = 0

Group like terms

x2 + 10xy2 - 3y  = -3

Subtract 3 from both sides

x2 + 10x52 + 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:  x2y2 - 5x + 4y  - 2 = 0
Put in
the Standard Equation form.

[Solution]

Graph the equation:  x2  + y - 4y  - 21 = 0

Solve

                           Step

x2  + y - 4y  - 21 = 0

Put in the Standard Equation form

x2  +  y- 4y  = 21

Add 21 to both sides

x2  +  y - 4y + 4 = 21 + 4

Take half the middle term and square it. 

x2  +  (y  - 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 + y - 7  = 0

[Solution]

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada