Lecture 10.8 - Equations of Circles

Lecture 10.8 - Equations of Circles

G²

Centered at the origin, we can draw a right triangle inside of a circle and use the coordinates x & y to write the pytahgorean theorem to solve for r (radius)

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Can we move the circle off of the origin?

​Circles on the Coordinate Plane

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Equation for a circle centered at the origin​

We're not in function Kansas anymore

Multiple Choice

Write the standard equation of the circle with the given center and radius; center (0,0) and radius: 3

• x²+y²=3
• x²+y²=6
• x²+y²=9

Multiple Choice

What is the equation of the circle?

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

• $$x^2+y^2=4$$

• $$\left(x+4\right)^2+\left(y+4\right)^2=16$$

• $$x^2+y^2=16$$

Multiple Select

Select all the equations for a circle with a diameter of 12.

• $$\left(x+8\right)^2+\left(y-4\right)^2=36$$

• $$\left(x-2\right)^2+\left(y+14\right)^2=144$$

• $$x^2+y^2=36$$

• $$x^2+y^2=144$$

• $$\left(x+6\right)^2+\left(y+6\right)^2=24$$

Imagine comparing quadratics to circles

Translating quadratic functions

Imagine comparing quadratics to circles

Translating quadratic functions

Multiple Choice

What equation represents the quadratic function?

• y = (x - 4)²
• y = (x + 4)²
• y = x² - 4
• y = x² + 4

Multiple Choice

What equation represents the quadratic function?

• y = (x - 4)²
• y = (x + 4)²
• y = x² - 4
• y = x² + 4

Equation for any circle

• r is the radius

• h is the x coordinate of the center

• k is the y coordinate of the center​

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​Essentially, you are shifting left/right/up/down relative to a circle centered at the origin

What do you think happens if there was a number outside of the set of parenthesis?

Multiple Choice

In the equation (x-3)²+(y-2)²=16, the center of the circle is...

• (3,2)
• (-3, -2)
• (-2, -3)
• (2, 3)

Multiple Choice

In the equation (x+2)²+(y-3)²=4, the radius of the circle is...

• 4
• 2
• 3
• 16

Multiple Choice

In the equation (x-4)²+(x-3)²=25, the center of the circle is at...

• (-3, -4)
• (4, 3)
• (-4, -3)
• (3, 4)

Multiple Choice

The diameter of a circle has length 12. The center is at (-5, 2). Give the equation of the circle.

• (x - 2)² + (y + 5)² = 36
• (x - 5)² + (y + 2)² = 6
• (x + 5)² + (y - 2)² = 36
• (x + 2)² + (y - 5)² = 6

Multiple Choice

Which graph corresponds to 
(x)²+ (y-6)²=9

• A
• B
• C
• D

Multiple Choice

Which graph corresponds to

(x-4)²+ (y+4)²=25

• A

• B

• C

• D

Draw

Graph the circle: $$9=\left(x-2\right)^2+\left(y+4\right)^2$$  

Draw

Graph the circle: $$16=\left(x+1\right)^2+\left(y-5\right)^2$$  

Multiple Choice

Find the equation of the circle with a center at (−8,3) that passes through the point (2,−5) that is located on the circle?

• (x + 8)² + (y − 3)² = 12.8

• (x + 3)² + (y − 8)² = 164

• x² + y ²= 12.8

• (x + 8)² + (y − 3)² = 164

Multiple Choice

Which is the equation of a circle given endpoints of the diameter are (0,5) (2,3)

• (x - 1)² + (y - 4)² = 2

• (x - 1)² + (y - 4)² = 4

• (x + 1)² + (y + 4)² = 1

• (x + 1)² + (y - 4)² = 4

Complete The Square

A square can simply not be completed without another square

Multiple Choice

When factoring
x² - 4x + 4 = 20,
what goes in the blank?
(x - __ )² = 20

• 4
• 2
• 8
• 20

Multiple Choice

Complete the square for
x² + 12x + ____

• x² + 12x + 144
• x² + 12x + 36
• x² + 12x - 36
• x² + +12x - 144

Multiple Choice

Complete the square.
x² + 6x + ____

• x² + 6x + 9
• x² + 6x - 36
• x² + 6x + 36
• x² + 6x - 9

Multiple Choice

Complete the square.
x² + 14x + ____

• x² + 14x - 196
• x² + 14x + 49
• x² + 14x - 49
• x² + 14x + 196

Multiple Choice

Complete the Square
x² + 10x + ____

• x² + 10x - 100
• x² + 10x + 100
• x² + 10x - 25
• x² + 10x + 25

Multiple Choice

Complete the Square
x² - 12x

• (x + 6)² + 12
• (x - 6)² - 24
• (x - 6)² - 12
• (x - 6)² - 36

Multiple Choice

Complete the Square
x² + 6x + 20

• (x + 3x)² + 11
• (x + 3)² + 11
• (x + 6)² + 11
• (x + 3)² + 29

Multiple Choice

Complete the Square
x² + 6x + 5

• (x + 3)² - 4
• (x + 6)² - 4
• (x + 3x)² - 4
• (x + 3)² + 9

Multiple Choice

What should be added to both sides of the following equation to complete the square?
x² + 12x = 5

• 12
• 6
• 36
• 5

Complete The Square... for a circle

A square can simply not be completed without another square

Multiple Choice

Determine the equation of the circle in standard form: x² + y² + 6x - 12y + 20 = 0

• (x - 3)² + (y + 6)² = 25

• (x + 3)² + (y - 6)² = 25

• (x - 3)² + (y + 6)² = 45

• (x + 3)² + (y - 6)² = 5

Multiple Choice

Determine the equation of the circle in standard form:

x² + y² + 4x + 6y - 36 = 0

• (x + 2)² + (y + 3)² = 49

• (x + 2)² + (y + 3)² = 13

• (x - 2)² + (y - 3)² = 49

• (x - 2)² + (y - 3)² = 13

Multiple Choice

Change the following equation to standard form

x² + y² - 4x - 5=0

• (x - 2)² + y² = 3

• (x - 2)² + y² = 9

• (x - 4)² + y²= 1

• (x - 4)² + (y - 1)²= 22

Multiple Choice

What are the solution(s) to the system of equations? (use graph paper) $$x^2+\left(y-2\right)^2=16$$   $$y-x=6$$  

• (0, -6)  (4, -2)

• (0, 6)  (4, 2)

• (-4, 2)  (0, 6)

• no solution

Multiple Choice

What are the solution(s) to the system of equations? (use graph paper) $$\left(x+4\right)^2+\left(y-1\right)^2=25$$   $$x=1$$  

• (1, 1)

• (-9, 1)  (1, 1)

• (-4, 6)

• (-4, 6)  (-4, -4)