You can write an equation of a circle in a coordinate plane if you know its radius and the coordinates of its center. In this topic, you will solve problems involving circles and the coordinate plane. In order to solve these problems, you will learn the following information:

• how to interpret the equation of a circle, (x − h)² + (y − k)² = r², to determine the radius and the coordinates of the center of a circle;
• how to write the standard equation of a circle when given the coordinates of the endpoints of its diameter or the coordinates of its center and a point on the circle; and
• how to identify a point on the circle when given the length of the radius and the coordinates of the center of the circle.
  

Essential Questions

• Given the radius and the coordinates of the center of a circle, how can a point on the circle be identified?
• How can you determine the radius and the coordinates of the center of a circle, when given the equation of the circle?
• How can you write the equation of a circle, when given the coordinates of its diameter?
• How can you write the equation of a circle, when given the coordinates of the center and a point on the circle?

 

Warm-Up

You have been getting horrible reception on your cell phone for the past two weeks. Your cell phone company has agreed to give you a “mini tower,” which you must install at home. As long as the cell phone is within close proximity of the “mini tower,” your reception will receive a boost. The grid shown represents the location of your home and the locations that you frequent. Keeping in mind that you must install the “mini tower” at home, will your cell phone receive a boost at each location?

Hover your cursor over the grid to view the range of the "mini tower." The range of the “mini tower” is represented by the area within the circle. You can see that the “mini tower” will fall short in a few locations.

Although you will receive a boost while at school and the gym, you will still have horrible reception while at church, the grocery store, and at work. Sadly, the "mini tower" will not be enough to fix your reception problem. Cell phone companies follow these steps when deciding on the placement of cell phone towers.  The companies must guarantee that their towers can cover a range that will satisfy their customers.