What type of graph is produced by the equation ? How can you turn this equation into graphing form in order to graph it?
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.
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.
Solution: The three sides of the triangle are , , and , where is the hypotenuse.
By the Pythagorean Theorem, .
Do all points on the circle from Example A satisfy the equation ?
Solution: 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.
Why does a circle with center and radius have the general equation ?
Solution: 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.
Concept Problem Revisited
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.
A circle is the set of all points equidistant from a given point. The general equation of a circle is .
1. What is the equation of a circle centered at with a radius of ?
2. What are the center and radius of the circle described by the equation:
3. Graph the circle from #2.
1. , , . 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.
2. Complete the square to rewrite the equation in graphing form:
The center of the circle is and the radius is 4.
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: