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.
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.
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.
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.
Graph the circle from #3.
Graph the following circles:
Find the center and radius of the circle described by each equation.
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:
To see the Review answers, open this PDF file and look for section 10.2.