Circle Equations

The Standard Equation of a Circle

Just the Standard Form, no foiling needed today

What is a Circle?

Infinite, coplanar points equidistant from a common center..

Circle Center

• Circles are named by their Center Point

• The Center of the Circle can be located using coordinates (h, k)

• All points that are r distance away from (h,k) are on the circle

Multiple Choice

Name the circle

• A

• BCD

• BC

• BACD

Multiple Choice

Name the circle:

• Circle B

• Circle DE

• Circle A

• Circle DBC

Multiple Choice

What is CB?

• diameter

• center

• circumference

• radius

Multiple Choice

The distance from the center of the circle to any point on the circle is called the...

• radius

• diameter

• circumference

• area

Multiple Choice

Which of the following is a radius of the circle?

• DA

• GE

• HI

• FG

Pythagorean Theorem

The Pythagorean Theorem describes all the points (x,y) that are distance r away from a center.

When the Center is at the Origin

The Equation for a Circle is just Pythagorean Theorem for a given radius

 $r^2=x^2+y^2$  

it describes all points (x,y) that satisfy the equation

                      $r^2=x^2+y^2$  

• There are 3 Variables related in this equation

• If you know the radius, you can choose one ordinate (x or y) to solve for the other and complete the ordered pair

• So this equation locates all the corrdinate pairs (x,y) that are r away from the center...

• ...for this version of the equation the Center is assumed to be at the origin (0,0)

Multiple Choice

What is the center of the circle with equation x² + y² = 1?

• (1, 1)

• (0, 0)

• not enough information

Multiple Choice

For the Circle 

 $4^2=x^2+y^2$  

 What is the radius of the following Circle?   hint:  we can see 

 $r^2=4^2$   so r= ?

• 4

• 16

• 2

• 8

Multiple Choice

 $x^2+y^2=16$  What is the radius of the following Circle? 

 hint,  $r^2=16$  

• 16

• 4

• 8

• not enough information

Multiple Select

Which of the following are correct equations of a circle with 

center (0, 0) and a radius of 3.

•  $x^2+y^2=9$ 

•  $x^2+y^2=3^2$ 

•  $3^2+y^2=r^2$ 

•  $x^2+3^2=r^2$ 

Multiple Choice

Write the standard equation of the circle with the given center and radius; center (0,0) and radius: 9

• x²+y²=9

• x²+y²=81

• x²+y²=3

Multiple Choice

What is the Radius?

• 16

• 4

• 8

Multiple Choice

What is the Radius?

• 3

• 4

• 5

• (3,4)

• 25

Multiple Choice

What is the Equation for this Circle? (hint, all points)

• $x^2+y^2=16$

• $3^2+4^2=5^2$
  

•  $x^2+y^2=25$  
  

•  $x^2+y^2=3^2$  
  

Multiple Choice

What is the Equation for this Circle?

• $x^2+y^2=16$

• $3^2+4^2=5^2$
  

•  $x^2+y^2=25$  
  

•  $x^2+y^2=3^2$  
  

When the Center isn't at the Origin

...then we translate the equation with a horizontal and vertical "shift" or slide

Circle Center (h,k)

When we translate a circle, for every point we can see that they move horizontal distance h and vertical distance y. 

We subtract : (x-h)  and (y-k) 
The equation for this circle is

 $3^2=\left(x-4\right)^2+\left(y-3\right)^2$  

Time to Explore...

open the links, try

https://bit.ly/3vAxoR4

https://bit.ly/3eexAzD

https://bit.ly/3uiXNmp

Remember

• the equation is for  $r^2$  not just r

• You SUBTRACT (h,k) in the equation

• ..so NEGATIVE shifts are ADDED in the equation

• (h,k) is the Center , it's also the shift

• h is the horizontal shift

• k is the vertical shift

• The equation gives all ordered pairs (x,y) for radius r from center (h,k)

Multiple Choice

What is the equation of a circle with radius 4 and center (0, 8)?

• x² + (y-8)² = 16

• (x-8)² + y² = 4

• x² + (y-8)² = 4

• x² + (y+8)² = 16

Multiple Choice

Write the equation of the circle with center C( -5 , 8 ) and radius = 7

• ( x - 5 )² + ( y + 8 )² = 14

• ( x - 5 )² + ( y - 8 )² = 49

• ( x - 5 )² + ( y + 8 )² = 49

• ( x + 5 )² + ( y - 8 )² = 49

Multiple Choice

What is the equation of the graph shown?

• $\left(x+1\right)^2+\left(y+1\right)^2=5$

• $\left(x-1\right)^2+\left(y-1\right)^2=5$

• $\left(x+1\right)^2+\left(y+1\right)^2=4$

• $\left(x+1\right)^2+\left(y+1\right)^2=2.24$

Multiple Choice

Which of the following graphs match the equation:

Multiple Choice

C is the center of the circle shown below. What is the equation of circle C?

Multiple Choice

Is the point (3 , 5) inside, outside or on the circle with equation x² + y² = 9?

• Inside the Circle

• Outside the Circle

• On the Circle

Multiple Choice

What is the equation of the circle, with centre (-4,5) and radius 6 units?

• (x - 4)^2 + (y + 5)^2 = 6

• (x - 4)^2 + (y + 5)^2 = 36

• (x + 4)^2 + (y - 5)^2 = 6

• (x + 4)^2 + (y - 5)^2 = 36

Multiple Choice

 

Which graph corresponds to

 $\left(x-8\right)^2+\left(y+2\right)^2=r^2$  

• A

• B

• C

• D

Multiple Choice

 

Which Equation Matches the Red Circle? 

•  $\left(x+4\right)^2+\left(y-4\right)^2=5^2$  
  

•  $\left(x-4\right)^2+\left(y-4\right)^2=5^2$  
  

•  $\left(x-4\right)^2+\left(y+4\right)^2=5^2$  
  

Multiple Choice

What is the equation of the circle?

• (x+1)² + (y +1)² = 3

• (x+1)² + (y -1)² = 3

• (x+1)² + (y +1)² = 9

• (x-1)² + (y -1)² = 9

Multiple Choice

What are the coordinates of the center and the radius of the circle with equation:

(x - 4)² + (y - 3)² = 25 ?

• Center (4,3)

  Radius = 5 units

• Center (4,3)

  Radius = 25 units

• Center (-4,-3)

  Radius = 25 units

• Center (-4,-3)

  Radius = 5 units

Multiple Choice

Write the equation of a circle with center (7, 0) with radius 3.

• (x - 7)² + y² = 9

• x² + (y -7)² = 9

• (x - 7)² + y² = 3

• x² + (y -7)² = 3

You have Finished! Now chill