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Equation of a circle – Demonstration

Posted in Different Math Theory topics, Math Theory by Lia on 09/05/2011

Let’s consider circle {{}{M}{}} with M(x_m, y_m) as circle’s center and {{}{r}{}} the circle’s radius.

The perimeter of circle {{}{M}{}} is the geometric locus for the point N(x, y), function of M(x_m, y_m)  and {{}{r}{}} . The circle’s equation is the expression that it is true for any N(x, y).

Using Pythagoras theorem for the right triangle {{}{MNP}{}} we have:

[1]  {{MP}^2} +{{PN}^2} ={{MN}^2} where:

[2]  MP = x-{x_m}

[3]  PN = y-{y_m}

[4]  MN = r

By replacing the values we have for the different segment lines ([2], [3], and [4]) into [1] we get the general form of circle’s equation:

[5]  tabular{11}{11}{{{{(x-{x_m})}^2} +{{(y-{y_m})}^2} = {r^2}}}

When the center of the circle M(x_m, y_m)  is coincident with the Origin of the Cartesian Coordinate System  ( O(0, 0) ), x_m = 0 and y_m = 0, [5] becomes:

[6]  tabular{11}{11}{{{x^2} +{y^2} ={r^2}}}

 

 

 

 

 

 

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