Recognizing Equations of Circles

New York State Common Core Math Geometry, Module 5, Lesson 18

Worksheets for Geometry, Module 5, Lesson 18

Student Outcomes

• Students complete the square in order to write the equation of a circle in center-radius form.
• Students recognize when a quadratic in x and y is the equation for a circle.

Recognizing Equations of Circles

Classwork

Opening Exercise

a. Express this as a trinomial: (𝑥 − 5)².
b. Express this as a trinomial: (𝑥 + 4)².
c. Factor the trinomial: 𝑥² + 12𝑥 +36.
d. Complete the square to solve the following equation: 𝑥² +6𝑥 = 40.

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Example 1

The following is the equation of a circle with radius 5 and center (1, 2). Do you see why?
𝑥² − 2𝑥 + 1 + 𝑦² − 4𝑦 + 4 = 25

Exercise 1

1. Rewrite the following equations in the form (𝑥 − 𝑎)² + (𝑦 − 𝑏)² = 𝑟².
   a. 𝑥² + 4𝑥 + 4 + 𝑦² − 6𝑥 + 9 = 36
   b. 𝑥² − 10𝑥 + 25 + 𝑦² + 14𝑦 + 49 = 4

Example 2

What is the center and radius of the following circle?
𝑥² + 4𝑥 + 𝑦² − 12𝑦 = 41

Exercises 2–4

2. Identify the center and radius for each of the following circles.
   a. 𝑥² − 20𝑥 + 𝑦² + 6𝑦 = 35
   b. 𝑥² − 3𝑥 + 𝑦² − 5𝑦 = 19/2
3. Could the circle with equation 𝑥² − 6𝑥 + 𝑦² − 7 = 0 have a radius of 4? Why or why not?
4. Stella says the equation 𝑥² −8𝑥 + 𝑦² + 2y = 5 has a center of (4,−1) and a radius of 5. Is she correct? Why or why not?

Example 3

Could 𝑥² + 𝑦² + 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 represent a circle?

Exercise 5

5. Identify the graphs of the following equations as a circle, a point, or an empty set.
   a. 𝑥² + 𝑦² +4𝑥 = 0
   b. 𝑥² + 𝑦² + 6𝑥 − 4𝑦 + 15 = 0
   c. 2𝑥² + 2𝑦² − 5𝑥 + 𝑦 + 13/4 = 0

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