LESSON : CIRCLES

Circles: Circles have the form ( x – h)² + ( y – k)² = r² ( h,k) represents the center of the circle r represents the radius of the circle Ex. ( x + 2)² + ( y – 1)² = 4 center ( -2, 1) radius = 2

Sometimes circles are not in the standard form and we have to manipulate the equation ( sometimes completing the square is helpful) Ex. x² + y² + 4x – 2y - 4 = 0 Regrouping and completing the squares we have: (x² + 4x + ___ ) + ( y² - 2y + ___) = 4 + ___ + ____ ( x² + 4x + 4) + ( y² - 2y + 1) = ( x + 2)² + ( y – 1)² = 9 So center ( -2, 1) radius = 3

Ex. Center is ( 2,2) and another point on the circle is ( 5,8) So we need get r² so we need to solve for it by plugging in the ( h, k) or the center and the ( x,y) or the point on the circle. ( 5 – 2)² + ( 8 – 2)² = r² 3² + 6² = r² 45 = r² So the equation would be ( x – 2)² + ( y- 2)² = 45

Graphing circles is fairly easy: 1. Plot the center of the circle first ( h,k) 2. Using the radius plot a point above, below, to the right, and to the left of the center point and sketch in the circle with those points. Examples on the board in class