P.3 Circles & Symmetry

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

Examples Check for symmetry with respect to both axes and the origin. xy2 + 10 = 0 Answer– X-axis symmetry only

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

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

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

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

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

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?

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

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?

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)=75+9+ 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.

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

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

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