Equation of a circle


The standard form of an equation of a circle is ( x - h )2 + ( y - k )2 = r2. The radius is r, the center of the circle is (h , k), and (x , y) is any point on the circle.



For example, suppose ( x - 2 )2 + ( y - 3 )2 = 42 is an equation of a circle. The center of this circle is located at ( 2 , 3 ) on the coordinate system and the radius is 4.


How to derive the standard form of an equation of a circle.

Start with the circle you see below. Then put the circle on the coordinate system.

Finally, label the circle to show the center and a point on the circle. 

Notice that (h, k) is the center of the circle and (x, y) is a point on the circle. The distance between (h, k) and (x, y) is the length of the radius.



To find the equation, we just need to use the distance formula which can be used to find the distance between two points.



  __________________
Distance formula = √  (x1 - x2)2 + (y1 - y2)2  


  __________________
d = √  (x1 - x2)2 + (y1 - y2)2  


Notice that the distance between (h,k) and (x,y) is r. Thus d = r

  __________________
r = √  (x1 - x2)2 + (y1 - y2)2  


Let (x1, y1) = (x,y) and (x2, y2) = (h,k)

  __________________
r = √  (x - h)2 + (y - k)2  


Square each side of the equation

r2 = ( x - h )2 + ( y - k )2




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