P.3 Circles & Symmetry. 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.

P.3 Circles& Symmetry

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

Check for symmetry with respect to both axes and the origin.

xy2 + 10 = 0

Answer– X-axis symmetry only

Check for symmetry with respect to both axes and the origin.

y = 9 – x2

Answer– Y-axis symmetry only

Check for symmetry with respect to both axes and the origin.

xy = 4

Answer– origin symmetry only

Equation of a Circle

Standard Form:

General Form:

(x – h)2 + (y – k) 2 = r2

Ax2 + Ay2 + Bx + Cy + D = 0

Standard Form:(x – h)2 + (y – k) 2 = r2

(h,k) = centerr = radius

(h,k)r

Determine the center and the radius of the following circles in standard form:

Equation Center

Radius

x2 + (y-5) 2 = 32

(x–1)2 + (y+7) 2 = 25

(x–2)2 + (y–3) 2 = 16

(x–1/3)2 + y 2 = 9/2

(x+6)2 +(x+2.3)2 =2.5

Determine the standard form of the following circles if given the center and the radius:

Equation Center

Radius

(-1,-5) r = 3

(3,4) r=8

(1.3,-6.5) r = 2.2

(0, -4) r =√ 3

(2,0) r = 5√ 5

General Form:Ax2 + Ay2 + Bx + Cy + D = 0

All circles in standard form can be easily converted to

general form…

how?

A,B,C & D are integers

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

Determine the center and the radius of the following circle in general form:

x2 + y2 - 6x – 8y – 75 = 0

What do you think?

x2 + y2 - 6x – 8y – 75 = 0Divide every term by “A”

Group x’s and y’s…Move “D” to the other side of the =

(x2 - 6x ) + (y2 – 8y )= 75

Complete the square of both groups…Remember, whatever you add to the left,

be sure to add to the right.

(x2 - 6x + 9)+(y2 – 8y +16)=75+9+ 16

Factor each group and simplify the right.

(x – 3)2+(y – 4)2 =100

Now you can identify the center and the 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 = 03x2 + 3y2 + 3x – 36y = 0

x2 + y2 – 14y – 1= 016x2 + 16y2 + 48x – 88y – 3 = 0

More ExamplesDetermine 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

More ExamplesDetermine 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