# 9.3 Circles

- Page ID
- 1034

A **circle** is the collection of points that are the same distance from a single point. What is the connection between the Pythagorean Theorem and a circle?

**Graphing Circles**

The single point that all the points on a circle are equidistant from is called the **center** of the circle. A circle does not have a focus or a directrix, instead it simply has a center. Circles can be recognized immediately from the general equation of a conic when the coefficients of \(x^{2}\) and \(y^{2}\) are the same sign and the same value. Circles are not functions because they do not pass the vertical line test. The distance from the center of a circle to the edge of the circle is called the **radius** of the circle. The distance from one end of the circle through the center to the other end of the circle is called the **diameter**. The diameter is equal to twice the radius.

The graphing form of a circle is:

\((x-h)^{2}+(y-k)^{2}=r^{2}\)

The center of the circle is at \((h, k)\) and the radius of the circle is \(r\). Note that this looks remarkably like the Pythagorean Theorem.

To graph a circle, first plot the center and then apply the radius. Take the following equation for a circle:

\((x-1)^{2}+(y+2)^{2}=9\)

The center is at (1,-2) . Plot that point and the four points that are exactly 3 units from the center.

## Examples

Example 1

Earlier, you were asked about the connection between circles and the Pythagorean Theorem. The reason why the graphing form of a circle looks like the Pythagorean Theorem is because each \(x\) and \(y\) coordinate along the outside of the circle forms a perfect right triangle with the radius as the hypotenuse.

Example 2

Graph the following conic: \((x+2)^{2}+(y-1)^{2}=1\)

Example 3

Turn the following equation into graphing form for a circle. Identify the center and the radius.

\(36 x^{2}+36 y^{2}-24 x+36 y-275=0\)

Complete the square and then divide by the coefficient of \(x^{2}\) and \(y^{2}\)

\(36 x^{2}-24 x+36 y^{2}+36 y=275\)

\(36\left(x^{2}-\frac{2}{3} x+\underline{-}\right)+36\left(y^{2}+y+\underline{x}\right)=275\)

\(36\left(x^{2}-\frac{2}{3} x+\frac{1}{9}\right)+36\left(y^{2}+y+\frac{1}{4}\right)=275+4+9\)

\(36\left(x-\frac{1}{3}\right)^{2}+36\left(y+\frac{1}{2}\right)^{2}=288\)

\(\left(x-\frac{1}{3}\right)^{2}+\left(y+\frac{1}{2}\right)^{2}=8\)

The center is \(\left(\frac{1}{3},-\frac{1}{2}\right)\). The radius is \(\sqrt{8}=2 \sqrt{2}\).

Example 4

Write the equation for the following circle.

\((x-1)^{2}+(y+2)^{2}=4\)

Example 5

Write the equation of the following circle.

The center of the circle is at (3,1) and the radius of the circle is \(r=4\). The equation is \((x-3)^{2}+(y-1)^{2}=16\)

Review

Graph the following conics:

1. \((x+4)^{2}+(y-3)^{2}=1\)

2. \((x-7)^{2}+(y+1)^{2}=4\)

3. \((y+2)^{2}+(x-1)^{2}=9\)

4. \(x^{2}+(y-5)^{2}=8\)

5. \((x-2)^{2}+y^{2}=16\)

Translate the following conics from standard form to graphing form.

6. \(x^{2}-4 x+y^{2}+10 y+18=0\)

7. \(x^{2}+2 x+y^{2}-8 y+1=0\)

8. \(x^{2}-6 x+y^{2}-4 y+12=0\)

9. \(x^{2}+2 x+y^{2}+14 y+25=0\)

10. \(x^{2}-2 x+y^{2}-2 y=0\)

Write the equations for the following circles.

11.

12.

13.

14.

15.