What type of graph is produced by the equation ? How can you turn this equation into graphing form in order to graph it?

### Equations of Circles

A **circle** is one example of a conic section. A circle is the set of all points that are equidistant from a center point. The general equation of a circle is where is the center of the circle and is the radius of the circle.

You can derive the equation of a circle with the help of the Pythagorean Theorem.

#### Deriving the Equation of a Circle

1. Label the three sides of the right triangle below in terms of , and . Then, use the Pythagorean Theorem to write an equation that shows the relationship between the three sides.

The three sides of the triangle are , , and , where is the hypotenuse.

By the Pythagorean Theorem, .

2. Do all points on the circle from the previous problem satisfy the equation ?

Yes, anywhere you move the point , the lengths of the sides of the triangle will still be , , and , so will still be true.

In fact, the set of points that make up the circle is exactly the set of points that satisfy the equation . This is why the general equation of a circle centered at the origin is , where is the radius of the circle.

3. Why does a circle with center and radius have the general equation ?

Pick a point on the circle and draw a right triangle connecting that point with the center of the circle.

The three sides of the triangle have lengths , and .

The relationship between the sides of the triangle is shown with the Pythagorean Theorem:

Points on the other side of the circle produce triangles with slightly different side lengths:

However, the relationship between the three sides of this triangle is the same, as shown below:

The set of points that satisfy the equation are the set of points that make up the circle.

**Examples**

**Example 1**

Earlier, you were asked how you can turn into graphing form in order to graph it.

The equation is the equation for a circle. You can tell it is a circle because there is both an term and a term, and the coefficients of each are the same. To turn this equation into the form , you must complete the square twice. Remember that when completing the square, you are trying to figure out what number to add to make a perfect square trinomial. In order to maintain the equality of the equation, you must add the same numbers to both sides of the equation.

#### Example 2

What is the equation of a circle centered at with a radius of ?

, , . The equation of the circle is . This can be simplified to . Notice that the signs of the 4 and the 5 in the equation are opposite from the signs of the 4 and the 5 in the center point.

#### Example 3

What are the center and radius of the circle described by the equation:

Complete the square to rewrite the equation in graphing form:

The center of the circle is and the radius is 4.

#### Example 4

Graph the circle from #3.

### Review

Graph the following circles:

1.

2.

3.

4.

5.

Find the center and radius of the circle described by each equation.

6.

7.

8.

9.

10.

11. Explain why a circle with radius that is centered at the origin will have the equation .

12. Use the Pythagorean Theorem to help explain why a circle with a center in the **second quadrant** with radius and center will have the equation . *Note that in the second quadrant the value for* *will be negative and the value for* *will be positive.*

13. Use the Pythagorean Theorem to help explain why a circle with a center in the **third quadrant** with radius and center will have the equation . *Note that in the third quadrant the value for * *will be negative and the value for* *will be negative.*

14. Write the equation of the following circle:

15. Write the equation of the following circle:

### Review (Answers)

To see the Review answers, open this PDF file and look for section 10.2.