Recognizing Equations of Circles

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

Worksheets for Geometry

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.

• Show Video

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

Check out our most popular games!
Fraction Concoction	Fact Family Game	Number Bond Garden
Online Addition Subtraction Game	Penguin Solitaire	Sawayama Solitaire
Ark Solitaire	Eldritch Invasion	Shenzhen Solitaire

Check out more Solitaire games here.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.