Solving Radical Equations

Radical Equations  Isolate the radical first  Undo the radical with a reciprocal power.

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

Determine the center and the radius of the following circles in standard form: EquationCenterRadius x 2 + (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 +(y+2.3) 2 =2.5

Determine the standard form of the following circles if given the center and the radius: EquationCenterRadius (-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: Ax 2 + Ay 2 + Bx + Cy + D = 0 All circles in standard form can be easily converted to general form… how ? A,B,C & D are integers

Standard FormGeneral 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: x 2 + y 2 - 6x – 8y – 75 = 0 What do you think?

x 2 + y 2 - 6x – 8y – 75 = 0 Divide every term by “A” Group x’s and y’s…Move “D” to the other side of the = (x 2 - 6x ) + (y 2 – 8y )= 75 Complete the square of both groups…Remember, whatever you add to the left, be sure to add to the right. (x 2 - 6x + 9)+(y 2 – 8y +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: EquationCenterRadius x 2 + y x – 6y – 4 = 0 2x 2 + 2y 2 + 8x + 20y + 10 = 0 3x 2 + 3y 2 + 3x – 36y = 0 x 2 + y 2 – 14y – 1= 0 16x y x – 88y – 3 = 0

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

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