Circles are generally much easier to work than parabolas. The only 2 important pieces of information in a circle are the center and the radius. The radius is the distance from the center of the circle to any point on the circle as the diagram illustrates.

 

 

Radius

 

 

 

 

 

The equation to use for all circles with radius, r, and a center at (h, k) is

(x – h)² + (y – k)² = r²

 

 

 

Example

Find the center and radius of the circle (x – 1)² + (y + 2)² = 9.

 

To find the center, look at the number that is with the x and use it for the x-coordinate, it is 1. To find the y-coordinate look at the number with the y it is +2. Since the original equation has a negative in that spot that means that the y-coordinate is –2.

 

Center = (1,-2)

 

Since 9 = r², set up an equation.

 

r² = 9

 

Solve for r.

 

r = 3

 

Answer!

 

 

 

 

Example

Find the center and radius of the circle x ² + (y - 3)² = 25.

 

To find the center, look at the number that is with the x and take it for the x-coordinate, in this case it would be 0 since there is not one. To find the y-coordinate look at the number with the y, it is -3. Since the original equation has a negative in that spot that means that the y-coordinate is opposite that sign making it +3. Therefore the coordinates for the center of the circle are (0,3).

 

Center = (0, 3)

 

Since 25 is r², set up an equation.

 

r² = 25

 

Solve for r.

 

r = 5

 

Answer!

 

 

 

Example

 

Graph x ² + 16x = -y² -12y

 

The first thing to realize with this problem is the fact that it is not in the correct form. Move all the variables on the left side of the equation by adding y² and 12y.

 

x ² + 16x + y² +12y = 0

 

Next use completing the square on x ² + 16x and y² +12y.

(x ² + 16x + 64) + (y² +12y + 36) = 0 + 64 + 36

 

Finish completing the square and simplify the terms on the right side.

 

(x + 8) ² + (y + 6)² = 100

 

Now it is in the correct form. The center is at (-8, -6) and has a radius of 10. Plot the point (-8, -6) and count out 10 units up, down, left, and right. Draw the circle from there.