7.1: Equations of Circles

7.1: Equations of Circles

Objective: Identify center and radius of a circle. Write an equation of a circle from a graph, and rewrite an equation in standard form.

Open Ended

What makes a circle unique?

• Think about the shape and properties of a circle!

Circle

• A collection of all points (x, y) that are equidistant from a fixed point (h, k)

• Radius ( r ) - the distance between the center and all points (x, y) on the circle

• Center: (h, k)

Standard From of a Circle

 $\left(x-h\right)^2+\left(y-k\right)^2=r^2$  

Center of the Circle: (h, k)
Radius Length =  $\sqrt{r^2}=r$  

Multiple Choice

What is the radius and center of

 $\left(x-9\right)^2+\left(y+3\right)^2=49$  

• (9, -3); r = 7

• (9, 3); r = 7

• (9, -3); r = 49

• (-9, -3); r = 7

Multiple Choice

What is the center and radius of

 $\left(x+1\right)^2+\left(y+2\right)^2=10$  

• (-1, -2); r = 10

• (-1, -2); r =  $\sqrt{10}$  

• (1, 2); r = 10

• (1, 2); r =  $\sqrt{10}$  

Multiple Choice

What is the center and radius of

 $x^2+y^2=9$  ?

• (3, 3); r = 3

• (0, 0); r = 9

• (0, 3); r = 3

• (0, 0); r = 3

Completing the Square

To get a Circle Equation in Standard form to identify the center and radius, we use completing the square.

Completing the Square with One Variable

 $x^2-18x+72=0$  

Multiple Choice

Complete the Square

 $x^2+6x+8=0$  

•  $\left(x-3\right)^2=-8$  

•  $\left(x+3\right)^2=17$  

•  $\left(x+3\right)^2=1$  

•  $\left(x-3\right)^2=8$  

Multiple Choice

Complete the square

 $y^2-12y+9=0$  

•  $\left(y+6\right)^2=9$  

•  $\left(y+6\right)^2=27$  

•  $\left(y-6\right)^2=45$  

•  $\left(y-6\right)^2=27$  

Completing the Square with two variables

Multiple Choice

What is the center of

 $x^2+y^2+6x-8y-11=0$  

• (6, 0)

• (-3, 4)

• (4, -3)

• (3, 4)

Multiple Choice

What is the radius of

 $x^2+y^2-2x+6y-10=0$  

•  $20$  

•  $0$  

•  $\sqrt{10}$  

•  $\sqrt{20}$  

Multiple Choice

What is the center of the circle whose endpoints of the diameter are (-4, 1) and (4, 5)

• (0, 0)

• (0, 3)

• (3, 0)

• (4, 5)