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Common Core For Geometry

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: 𝑥^{2} + 12𝑥 +36.

d. Complete the square to solve the following equation: 𝑥^{2} +6𝑥 = 40.

**Example 1**

The following is the equation of a circle with radius 5 and center (1, 2). Do you see why?

𝑥^{2} − 2𝑥 + 1 + 𝑦^{2} − 4𝑦 + 4 = 25

**Exercise 1**

- Rewrite the following equations in the form (𝑥 − 𝑎)
^{2}+ (𝑦 − 𝑏)^{2}= 𝑟^{2}.

a. 𝑥^{2}+ 4𝑥 + 4 + 𝑦^{2}− 6𝑥 + 9 = 36

b. 𝑥^{2}− 10𝑥 + 25 + 𝑦^{2}+ 14𝑦 + 49 = 4

**Example 2**

What is the center and radius of the following circle?

𝑥^{2} + 4𝑥 + 𝑦^{2} − 12𝑦 = 41

**Exercises 2–4**

- Identify the center and radius for each of the following circles.

a. 𝑥^{2}− 20𝑥 + 𝑦^{2}+ 6𝑦 = 35

b. 𝑥^{2}− 3𝑥 + 𝑦^{2}− 5𝑦 = 19/2 - Could the circle with equation 𝑥
^{2}− 6𝑥 + 𝑦^{2}− 7 = 0 have a radius of 4? Why or why not? - Stella says the equation 𝑥
^{2}−8𝑥 + 𝑦^{2}+ 2y = 5 has a center of (4,−1) and a radius of 5. Is she correct? Why or why not?

**Example 3**

Could 𝑥^{2} + 𝑦^{2} + 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 represent a circle?

**Exercise 5**

- Identify the graphs of the following equations as a circle, a point, or an empty set.

a. 𝑥^{2}+ 𝑦^{2}+4𝑥 = 0

b. 𝑥^{2}+ 𝑦^{2}+ 6𝑥 − 4𝑦 + 15 = 0

c. 2𝑥^{2}+ 2𝑦^{2}− 5𝑥 + 𝑦 + 13/4 = 0

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