PRE-CAL.pdf

Why conic sections? Architecture Conics in Real life Cathedral de la Canton Tower Resurrection Guangdong Evry , France. Province, China.  Architecture Conics in Real life Sydney Harbor Bridge, Sydney , National Radio...

Why conic sections? Architecture Conics in Real life Cathedral de la Canton Tower Resurrection Guangdong Evry , France. Province, China.  Architecture Conics in Real life Sydney Harbor Bridge, Sydney , National Radio Australia Astronomy Observatory Charlottesville, Virginia, USA. Architecture Conics in Real life Lyceum of the Philippines University Cavite Campus (x – h)2 + (y - k)2 = r2 x2 + y2 = r2 (center is not in the origin) (center is the origin) where (h, k) is the center and r is the radius Ax2 + By2 + Cx + Dy + E = 0  General form: Ax2 + By2 + Cx + Dy + E = 0 Determine the general form of the equation of a circle: (1) Center at the origin, radius 2 x2 + y2 = r2 standard form of the equation x2 + y2 = 22 substitute the given x2 + y2 = 4 simplify x2 + y2 -4 = 0 General form Determine the general form of the equation of a circle: (2) Center (-1, 2) radius = 𝟑 (x - h)2 + (y - k)2 = r2 standard form of the equation (x + 1)2 + (y-2) 2 = ( 𝟑)2 substitute the given x2 +2x +1 + y2 -4y +4 = 3 simplify x2 + y2 + 2x - 4y +1 +4 - 3 = 0 x2 + y2 +2x -4y +2 = 0 General form Ax2 + By2 + Cx + Dy + E = 0 Given the General Form of Equation of Circle, identify its Center and Radius. Example 1. x2 + y2 -8x + 7 = 0 (x – h)2 + (y - k)2 = r2 x2 -8x + y2 = -7 (x2 – 8x + 16) + y2 = -7 + 16 Completing the Square (x - 4 )2 + y2 = 9 C = (4, 0) r=3 Given the General Form of Equation of Circle, identify its Center and Radius. Example 2. 4x2 + 4y2 + 8x - 8y – 4 = 0 (x – h)2 + (y - k)2 = r2 4x2 + 8x + 4y2– 8y = 4 4(x2 + 2x) + 4(y2– 2y) = 4 4(x2 + 2x+1) + 4(y2– 2y+1) = 4 +1 (4) +1 (4) 4(x+1)2 + 4(y-1)2 = 12 (x+1)2 + (y-1)2 = 3 C = (-1, 1) r = 1.73