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Math 114 Notes : Circle Equation The DISTANCE FORMULA is d ( x2 x1 ) 2 ( y2 y1 ) 2 where the two points we want to find the distance between is ( x1 , y1 ) and ( x2 , y2 ) . Suppose we change the variables in the distance formula so that we have r ( x h) 2 ( y k ) 2 . r 2 ( x h)2 ( y k )2 . An example of this equation is 52 ( x 0)2 ( y 0) 2 . Below is a table of values (points) for this equation and a graph. Now, square both sides. We get The graph is a CIRCLE with center (0, 0) and radius 5 (count the tic marks on the x-axis to see that the radius is 5). We started with the equation r 2 ( x h)2 ( y k )2 . The radius is r and the center is (h, k). CIRCLE EQUATION: r 2 ( x h)2 ( y k )2 radius = r center = (h, k) Find the center and radius of the following circles using the circle equation. 1. ( x 5) 2 ( y 6) 2 49 2. x 2 y 2 36 (h int : ( x 0) 2 ( y 0) 2 36) 3. ( x 2) 2 ( y 3) 2 1 For #1, For #2, For #3, ( x 5) 2 ( y 6) 2 49 ( x h) ( y k ) r ( x 0) 2 ( y 0) 2 36 2 2 2 ( x h) ( y k ) r ( x 2) 2 ( y 3) 2 1 2 ( x h) ( y k ) r 2 2 2 2 2 so h = 5, k = 6 and r 2 49 . Center is (5, 6); Radius is 7. so h = 0, k = 0 and r 2 36 . Center is (0, 0); Radius is 6 so h = 2, k = - 3 and r 2 1 . Center = (2, -3); Radius = 1.