1. P.3 Circles
& Symmetry

2. Symmetry Symmetric to the x-axis Symmetric to the origin
Symmetric to the y-axis
Replace all of the x with –x
Symmetric to the x-axis
Replace all of the y with –y
Symmetric to the origin
Replace all of the y with –y and x with -x

3. Examples Check for symmetry with respect to both axes and the origin.
xy = 0
Answer– X-axis symmetry only

4. Examples Check for symmetry with respect to both axes and the origin.
y = 9 – x2
Answer– Y-axis symmetry only

5. Examples Check for symmetry with respect to both axes and the origin.
xy = 4
Answer– origin symmetry only

6. Equation of a Circle Standard Form: General Form:
(x – h)2 + (y – k) 2 = r2
General Form:
Ax2 + Ay2 + Bx + Cy + D = 0

7. Standard Form: (x – h)2 + (y – k) 2 = r2 (h,k) = center r = radius

8. Equation Center Radius
Determine the center and the radius of the following circles in standard form:
Equation
Center
Radius
(x–2)2 + (y–3) 2 = 16
(x–1)2 + (y+7) 2 = 25
x2 + (y-5) 2 = 32
(x–1/3)2 + y 2 = 9/2
(x+6)2 +(x+2.3)2 =2.5

9. Equation Center Radius
Determine the standard form of the following circles if given the center and the radius:
Equation
Center
Radius
(3,4) r=8
(-1,-5) r = 3
(0, -4) r =√ 3
(2,0) r = 5√ 5
(1.3,-6.5) r = 2.2

10. All circles in standard form can be easily converted to
General Form:
Ax2 + Ay2 + Bx + Cy + D = 0
A,B,C & D are integers
All circles in standard form can be easily converted to
general form…
how?

11. Standard Form General Form
(x–2)2 + (y–3) 2 = 16
(x–1)2 + (y+7) 2 = 25
(x–1/3)2 + y 2 = 9/2
(x+6)2 +(y+2.3)2 =2.5

12. What do you think? x2 + y2 - 6x – 8y – 75 = 0
Determine the center and the radius of the following circle in general form:
x2 + y2 - 6x – 8y – 75 = 0
What do you think?

13. x2 + y2 - 6x – 8y – 75 = 0 (x2 - 6x ) + (y2 – 8y )= 75
Divide every term by “A”
(x2 - 6x ) + (y2 – 8y )= 75
Group x’s and y’s…Move “D” to the other side of the =
(x2 - 6x + 9)+(y2 – 8y +16)=
Complete the square of both groups…Remember, whatever you add to the left, be sure to add to the right.
(x – 3)2+(y – 4)2 =100
Now you can identify the center and the radius…
Factor each group and simplify the right.

14. Equation Center Radius
Determine the center and the radius of the following circles in general form:
Equation
Center
Radius
x2 + y2 + 12x – 6y – 4 = 0
2x2 + 2y2 + 8x + 20y + 10 = 0
3x2 + 3y2 + 3x – 36y = 0
x2 + y2 – 14y – 1= 0
16x2 + 16y2 + 48x – 88y – 3 = 0

15. passing through the point (2,7)
More Examples
Determine the standard form and general form of the following circle:
Center = (5,3)
passing through the point (2,7)
(5,3)
(2,7)
Picture not drawn to scale

16. Endpoints of the diameter :
More Examples
Determine the standard form and general form of the following circle:
Endpoints of the diameter :
(4,6) and (-8,1)
(-8,1)
(4,6)
Picture not drawn to scale