# 9.7: Extension: Writing and Graphing the Equations of Circles

**At Grade**Created by: CK-12

## Learning Objectives

- Graph a circle.
- Find the equation of a circle in the \begin{align*}x-y\end{align*} plane.
- Find the radius and center, given the equation of a circle and vice versa.
- Find the equation of a circle, given the center and a point on the circle.

## Graphing a Circle in the Coordinate Plane

Recall that the definition of a circle is the set of all points that are the same distance from the center. This definition can be used to find an equation of a circle in the coordinate plane.

Let’s start with the circle centered at (0, 0). If \begin{align*}(x, y)\end{align*} is a point on the circle, then the distance from the center to this point would be the radius, \begin{align*}r\end{align*}. \begin{align*}x\end{align*} is the horizontal distance \begin{align*}y\end{align*} is the vertical distance. This forms a right triangle. From the Pythagorean Theorem, the equation of a circle, *centered at the origin* is \begin{align*}x^2+y^2=r^2\end{align*}.

**Example 1:** Graph \begin{align*}x^2+y^2=9\end{align*}.

**Solution:** The center is (0, 0). It’s radius is the square root of 9, or 3. Plot the center, and then go out 3 units in every direction and connect them to form a circle.

The center does not always have to be on (0, 0). If it is not, then we label the center \begin{align*}(h, k)\end{align*} and would use the distance formula to find the length of the radius.

\begin{align*}r=\sqrt{(x-h)^2+(y-k)^2}\end{align*}

If you square both sides of this equation, then we would have the standard equation of a circle.

**Standard Equation of a Circle:** The standard equation of a circle with center \begin{align*}(h, k)\end{align*} and radius \begin{align*}r\end{align*} is \begin{align*}r^2=(x-h)^2+(y-k)^2\end{align*}.

**Example 2:** Find the center and radius of the following circles.

a) \begin{align*}(x-3)^2+(y-1)^2=25\end{align*}

b) \begin{align*}(x+2)^2+(y-5)^2=49\end{align*}

**Solution:**

a) Rewrite the equation as \begin{align*}(x-3)^2+(y-1)^2=5^2\end{align*}. The center is (3, 1) and \begin{align*}r = 5\end{align*}.

b) Rewrite the equation as \begin{align*}(x-(-2))^2+(y-5)^2=7^2\end{align*}. The center is (-2, 5) and \begin{align*}r = 7\end{align*}.

When finding the center of a circle always take the ** opposite sign** of what the value is in the equation.

**Example 3:** Find the equation of the circle below.

**Solution:** First locate the center. Draw in the horizontal and vertical diameters to see where they intersect.

From this, we see that the center is (-3, 3). If we count the units from the center to the circle on either of these diameters, we find \begin{align*}r = 6\end{align*}. Plugging this into the equation of a circle, we get: \begin{align*}(x-(-3))^2+(y-3)^2=6^2\end{align*} or \begin{align*}(x+3)^2+(y-3)^2=36\end{align*}.

## Finding the Equation of a Circle

**Example 4:** Determine if the following points are on \begin{align*}(x+1)^2+(y-5)^2=50\end{align*}.

a) (8, -3)

b) (-2, -2)

**Solution:** Plug in the points for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} in \begin{align*}(x+1)^2+(y-5)^2=50\end{align*}.

a) \begin{align*}(8+1)^2+(-3-5)^2=50\!\\ 9^2+(-8)^2=50\!\\ 81+64 \ne 50\end{align*}

(8, -3) is *not* on the circle

b) \begin{align*}(-2+1)^2+(-2-5)^2=50\!\\ (-1)^2+(-7)^2=50\!\\ 1+49=50\end{align*}

(-2, -2) is on the circle

**Example 5:** Find the equation of the circle with center (4, -1) and passes through (-1, 2).

**Solution:** First plug in the center to the standard equation.

\begin{align*}(x-4)^2+(y-(-1))^2&=r^2 \\ (x-4)^2+(y+1)^2&=r^2\end{align*}

Now, plug in (-1, 2) for \begin{align*}x\end{align*} and \begin{align*}y\end{align*} and solve for \begin{align*}r\end{align*}.

\begin{align*}(-1-4)^2+(2+1)^2&=r^2\\ (-5)^2+(3)^2&=r^2\\ 25+9&=r^2\\ 34&=r^2\end{align*}

Substituting in 34 for \begin{align*}r^2\end{align*}, the equation is \begin{align*}(x-4)^2+(y+1)^2=34\end{align*}.

## Review Questions

- Questions 1-4 are similar to Examples 1 and 2.
- Questions 5-8 are similar to Example 3.
- Questions 9-11 are similar to Example 4.
- Questions 12-15 are similar to Example 5.

Find the center and radius of each circle. Then, graph each circle.

- \begin{align*}(x+5)^2+(y-3)^2=16\end{align*}
- \begin{align*}x^2+(y+8)^2=4\end{align*}
- \begin{align*}(x-7)^2+(y-10)^2=20\end{align*}
- \begin{align*}(x+2)^2+y^2=8\end{align*}

Find the equation of the circles below.

- Is (-7, 3) on \begin{align*}(x+1)^2+(y-6)^2=45\end{align*}?
- Is (9, -1) on \begin{align*}(x-2)^2+(y-2)^2=60\end{align*}?
- Is (-4, -3) on \begin{align*}(x+3)^2+(y-3)^2=37\end{align*}?
- Is (5, -3) on \begin{align*}(x+1)^2+(y-6)^2=45\end{align*}?

Find the equation of the circle with the given center and point on the circle.

- center: (2, 3), point: (-4, -1)
- center: (10, 0), point: (5, 2)
- center: (-3, 8), point: (7, -2)
- center: (6, -6), point: (-9, 4)