# 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.

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

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

1. 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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