### Using equations of circles

Now that we know what the equation of a circle means, we can use it to identify the center and the radius and sketch the graph of the circle in the plane.  But just for a refresher, let’s restate the definition of the equation of a circle.

DEFINITION:  The equation of a circle with center $$\left( {h,k} \right)$$ and radius $$r$$ is given by

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

Let’s implement this information now.

EXAMPLE:  Identify the center and radius of the circle.  Then sketch the graph of the circle.

$${\left( {x - \frac{1}{2}} \right)^2} + {\left( {y + \frac{1}{2}} \right)^2} = 4$$

SOLUTION:  We manipulate this equation slightly to get it exactly in the form above:

$${\left( {x - \frac{1}{2}} \right)^2} + {\left( {y + \frac{1}{2}} \right)^2} = 4$$

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

Now this equation is in the general form.  From here we can conclude that

$${\text{Radius}} = 2$$

$${\text{Centeris}}\left( {\Large \frac{1}{2}, - \Large \frac{1}{2}} \right)$$

Here is the graph: EXAMPLE:  Identify the center and radius of the circle.  Then sketch the graph of the circle.

$${\left( {x - 2} \right)^2} + {\left( {y - \frac{7}{2}} \right)^2} = 7$$

SOLUTION:  Again we manipulate the equation into the general form.  We have

$${\left( {x - 2} \right)^2} + {\left( {y - \frac{7}{2}} \right)^2} = 7$$

$${\left( {x - 2} \right)^2} + {\left( {y - \frac{7}{2}} \right)^2} = {\left( {\sqrt 7 } \right)^2}$$

Then the radius of the circle is $$\sqrt 7 \approx 2.6$$, and the center is $$\left( {2,\frac{7}{2}} \right)$$. The graph is below. Below you can download some free math worksheets and practice. 2062 x

Graph each equation.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question: Watch bellow how to solve this example: 1952 x

Identify the center and radius of each. Then sketch the graph.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question: Watch bellow how to solve this example: 1826 x

Identify the center and radius of each. Then sketch the graph.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question: Watch bellow how to solve this example:

### Geometry

Circles
Congruent Triangles
Constructions
Parallel Lines and the Coordinate Plane
Properties of Triangles

### Algebra and Pre-Algebra

Beginning Algebra
Beginning Trigonometry
Equations
Exponents
Factoring
Linear Equations and Inequalities
Percents
Polynomials