You draw a circle that is centered at the origin. You measure the diameter of the circle to be 32 units. Does the point

### Circles Centered at the Origin

Until now, your only reference to circles was from geometry. A **circle** is the set of points that are equidistant (the **radius**) from a given point (the **center**). A line segment that passes through the center and has endpoints on the circle is a **diameter**.

Now, we will take a circle and place it on the

#### Finding the Equation of a Circle

Step 1: On a piece of graph paper, draw an

Step 2: Using the length of each side of the right triangle, show that the Pythagorean Theorem is true.

Step 3: Now, instead of using

The **equation of a circle**, centered at the origin, is

Let's find the radius of

To find the radius, we can set

Now, let's find the equation of the circle with center at the origin and passes through

Using the equation of the circle, we have:

So, the equation is

Finally, let's determine if the point

Substitute the point in for

The point is not on the circle.

### Examples

#### Example 1

Earlier, you were asked to determine if the point

From this lesson, you know that the equation of a circle that is centered at the origin is

With the point

Plug these values into the equation of the circle. If they result in a true statement, the point lies on the circle.

Therefore the point does not lie on the circle.

#### Example 2

Graph and find the radius of

#### Example 3

Find the equation of the circle with a radius of

Plug in

#### Example 4

Find the equation of the circle that passes through

Plug in

The equation is

#### Example 5

Determine if

Plug in

\begin{align*}(-10)^2+7^2&=149 \\ 100+49&=149\end{align*}

Yes, the point is on the circle.

### Review

Graph the following circles and find the radius.

- \begin{align*}x^2+y^2=9\end{align*}
- \begin{align*}x^2+y^2=64\end{align*}
- \begin{align*}x^2+y^2=8\end{align*}
- \begin{align*}x^2+y^2=50\end{align*}
- \begin{align*}2x^2+2y^2=162\end{align*}
- \begin{align*}5x^2+5y^2=150\end{align*}

Write the equation of the circle with the given radius and centered at the origin.

- 14
- 6
- \begin{align*}9 \sqrt{2}\end{align*}

Write the equation of the circle that passes through the given point and is centered at the origin.

- \begin{align*}(7,-24)\end{align*}
- \begin{align*}(2,2)\end{align*}
- \begin{align*}(-9,-10)\end{align*}

Determine if the following points are on the circle, \begin{align*}x^2+y^2=74\end{align*}.

- \begin{align*}(-8,0)\end{align*}
- \begin{align*}(7,-5)\end{align*}
- \begin{align*}(6,-6)\end{align*}

**Challenge** In Geometry, you learned about tangent lines to a circle. Recall that the tangent line touches a circle at one point and is perpendicular to the radius at that point, called the point of tangency.

- The equation of a circle is \begin{align*}x^2+y^2=10\end{align*} with point of tangency \begin{align*}(-3, 1)\end{align*}.
- Repeat the steps in #16 to find the equation of the tangent line to \begin{align*}x^2+y^2=34\end{align*} with a point of tangency of \begin{align*}(3, 5)\end{align*}.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 10.3.