Equation of a Circle

Equation of a Circle

Learning Target: I can write the equation of a circle and graph a circle from it's equation.

Multiple Choice

What does the equation of a circle with center at (0,0) look like?

• $y=x^2$

• $y=mx+b$

• $x^2+y^2=r^2$

• $y=a\left(x-h\right)^2+k$

Equation of a circle with center at (0,0)

The equation we use for a circle with the center at (0,0)  is

 $x^2+y^2=r^2$ 



the r stands for the radius of the circle.  

Multiple Choice

What is the equation for this circle?

• (x+1)+(y+1)=9
• x²+y²=9
• x+y=9

Multiple Choice

Write the equation at the origin with radius 6.  

• x^2 + y^2 = 6
• x^2 + y^2 = 36
• x  + y = 6
• x + y = 36

Multiple Choice

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

• (1, 1)
• (0, 0)
• not enough information
• (-1, -1)

What if the center isn't (0,0)?

Think back to tranformations of parabolas, we used the point (h,k) to talk about our vertex.

We can use the same point to talk about the center of our circle.

Equation of a circle

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

Here (h,k) is the center of the circle and r is the raidus.

Open Ended

Why has there not been and h and k value in the equations we have looked at so far, like this one?

 $x^2+y^2=81$  

Writing Equations of Circles

We can write equations of circles from their graphs.

Locate the center of the circle. This usually marked with a dot

Multiple Choice

What is the center of this circle?

• (2, 2)

• (2, -2)

• (0, 0)

• (-2, 2)

Writing Equations of Circles

The other piece of information we need is the radius.

You can calculate the radius by counting the number of units from the center to any point on the circle.

Multiple Choice

What is the radius of this circle?

• 2

• 6

• 3

• 5

Writing Equations of Circles

 Now we can place our values into our equation

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

Then simplify

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

Notice that the coordinates for the center (h,k) have the opposite sign in the equation. Our (h,k) was (-2, 2), but in the final equation it was

 $\left(x\ +2\right)\ ^2\ \ \ \ \ and\ \ \ \ \ \ \ \left(y-2\right)^2$  

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 is the center and radius for a circle with the equation:
(x-4)²+(y+2)²=25

• C:(4, -2), r=5
• C:(-4, 2), r=5
• C:(4, -2), r=25
• C:(-4, 2), r=25

Multiple Choice

What is the radius of the circle
(x+7)²+(y-2)²=144?

• 7
• 12
• 13

Multiple Choice

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

• (x - ^7)^2 + y^2 = 9
• x^2 + (y -7)^2 = 9
• (x - 7)^2 + y^2 = 3
• x^2 + (y -7)^2 = 3

Multiple Choice

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

• A
• B
• C
• D