Writing the Equation for a Circle

New York State Common Core Math Geometry, Module 5, Lesson 17

Worksheets for Geometry

Student Outcomes

• Students write the equation for a circle in center-radius form, (x - a)² + (y - b)² = r² using the Pythagorean theorem or the distance formula.
• Students write the equation of a circle given the center and radius. Students identify the center and radius of a circle given the equation.

Writing the Equation for a Circle

Classwork

Exercises 1–2

1. What is the length of the segment shown on the coordinate plane below?
2. Use the distance formula to determine the distance between points (9, 15) and (3, 7)

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Example 1

If we graph all of the points whose distance from the origin is equal to 5, what shape will be formed?

Example 2

Let’s look at another circle, one whose center is not at the origin. Shown below is a circle with center (2, 3) and radius 5.

Exercises 3–11

3. Write an equation for the circle whose center is at (9, 0) and has radius 7.
4. Write an equation whose graph is the circle below.
5. What is the radius and center of the circle given by the equation (𝑥 + 12)² + (𝑦 − 4)² = 81?
6. Petra is given the equation (𝑥 − 15)² + (𝑦 + 4)² = 100 and identifies its graph as a circle whose center is (−15, 4) and radius is 10. Has Petra made a mistake? Explain.
7. a. What is the radius of the circle with center (3, 10) that passes through (12, 12)?
   b. What is the equation of this circle?
8. A circle with center (2,−5) is tangent to the 𝑥-axis.
   a. What is the radius of the circle?
   b. What is the equation of the circle?
9. Two points in the plane, 𝐴(−3, 8) and 𝐵(17, 8), represent the endpoints of the diameter of a circle.
   a. What is the center of the circle? Explain.
   b. What is the radius of the circle? Explain.
   c. Write the equation of the circle.
10. Consider the circles with the following equations:
    𝑥² + 𝑦² = 25 and (𝑥 − 9)² + (𝑦 − 12)² = 100.
    a. What are the radii of the circles?
    b. What is the distance between the centers of the circles?
    c. Make a rough sketch of the two circles to explain why the circles must be tangent to one another.
11. A circle is given by the equation (𝑥² + 2𝑥 + 1)+ (𝑦² + 4𝑦 + 4) = 121.
    a. What is the center of the circle?
    b. What is the radius of the circle?
    c. Describe what you had to do in order to determine the center and the radius of the circle

Lesson Summary

(𝑥 − 𝑎)² +(𝑦 − 𝑏)² = 𝑟² is the center-radius form of the general equation for any circle with radius 𝑟 and center (𝑎, 𝑏).

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