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.
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