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First we need to find the distance between two points which is the diameter of the circle .
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Suppose :
a = ( - 4 , 8 )
b = ( 4 , - 4 )
We have following equation to find the distance between two points.
![distance = diameter = d](https://tex.z-dn.net/?f=distance%20%3D%20diameter%20%3D%20d)
![d = \sqrt{ ( \: {y(b) - y(a)} \: )^{2} + ( \: {x(b) - x(a)} \: )^{2} } \\](https://tex.z-dn.net/?f=d%20%3D%20%20%5Csqrt%7B%20%28%20%5C%3A%20%7By%28b%29%20-%20y%28a%29%7D%20%5C%3A%20%29%5E%7B2%7D%20%2B%20%20%28%20%5C%3A%20%7Bx%28b%29%20-%20x%28a%29%7D%20%5C%3A%20%29%5E%7B2%7D%20%20%7D%20%20%5C%5C%20)
Now just need to put the coordinates :
![d = \sqrt{ ({ - 4 - 8})^{2} + ( \: {4 - ( - 4)} \: )^{2} } \\](https://tex.z-dn.net/?f=d%20%3D%20%20%5Csqrt%7B%20%28%7B%20-%204%20-%208%7D%29%5E%7B2%7D%20%2B%20%20%28%20%5C%3A%20%7B4%20-%20%28%20-%204%29%7D%20%5C%3A%20%29%5E%7B2%7D%20%20%7D%20%20%5C%5C%20)
![d = \sqrt{ ({ - 12})^{2} + ({4 + 4})^{2} }](https://tex.z-dn.net/?f=d%20%3D%20%20%5Csqrt%7B%20%28%7B%20-%2012%7D%29%5E%7B2%7D%20%2B%20%20%28%7B4%20%2B%204%7D%29%5E%7B2%7D%20%20%7D%20)
![d = \sqrt{ ({ - 12})^{2} + ({8})^{2} }](https://tex.z-dn.net/?f=d%20%3D%20%20%5Csqrt%7B%20%28%7B%20-%2012%7D%29%5E%7B2%7D%20%2B%20%20%28%7B8%7D%29%5E%7B2%7D%20%20%7D%20)
![d = \sqrt{144 + 64}](https://tex.z-dn.net/?f=d%20%3D%20%20%5Csqrt%7B144%20%2B%2064%7D%20)
![d = \sqrt{208}](https://tex.z-dn.net/?f=d%20%3D%20%20%5Csqrt%7B208%7D%20)
![d = \sqrt{4 \times 52}](https://tex.z-dn.net/?f=d%20%3D%20%20%5Csqrt%7B4%20%5Ctimes%2052%7D%20)
![d = \sqrt{4 \times 4 \times 13}](https://tex.z-dn.net/?f=d%20%3D%20%20%5Csqrt%7B4%20%5Ctimes%204%20%5Ctimes%2013%7D%20)
![d = \sqrt{ {4}^{2} \times 13 }](https://tex.z-dn.net/?f=d%20%3D%20%20%5Csqrt%7B%20%7B4%7D%5E%7B2%7D%20%5Ctimes%2013%20%7D%20)
![d = 4 \sqrt{13}](https://tex.z-dn.net/?f=d%20%3D%204%20%5Csqrt%7B13%7D%20)
This is the diameter of the circle.
We know that :
![diameter = 2 \times radius](https://tex.z-dn.net/?f=diameter%20%3D%202%20%5Ctimes%20radius)
So :
![4 \sqrt{13} = 2 \times radius](https://tex.z-dn.net/?f=4%20%5Csqrt%7B13%7D%20%20%3D%202%20%5Ctimes%20radius)
Divide sides by 2
![\frac{4 \sqrt{13} }{2} = \frac{2 \times radius}{2} \\](https://tex.z-dn.net/?f=%20%5Cfrac%7B4%20%5Csqrt%7B13%7D%20%7D%7B2%7D%20%20%3D%20%20%5Cfrac%7B2%20%5Ctimes%20radius%7D%7B2%7D%20%20%5C%5C%20)
![2 \sqrt{13} = radius](https://tex.z-dn.net/?f=2%20%5Csqrt%7B13%7D%20%20%3D%20radius)
Remember it.
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To find the equation of the circle except the radius , we need the coordinates of the centre.
<em>The</em><em> </em><em>mid</em><em>point</em><em> of</em><em> the</em><em> </em><em>dia</em><em>meter</em><em> is</em><em> the</em><em> </em><em>ce</em><em>ntre</em><em> of</em><em> the</em><em> </em><em>ci</em><em>rcle</em><em> </em><em>.</em>
So we have to find the midpoint of the diameter .
Look :
![midpoint = center = o](https://tex.z-dn.net/?f=midpoint%20%3D%20center%20%3D%20o)
![o = ( \: \frac{x(a) + x(b)}{2} \: , \: \frac{y(a) + y(b)}{2} \: ) \\](https://tex.z-dn.net/?f=o%20%3D%20%28%20%20%5C%3A%20%5Cfrac%7Bx%28a%29%20%2B%20x%28b%29%7D%7B2%7D%20%20%5C%3A%20%2C%20%5C%3A%20%20%5Cfrac%7By%28a%29%20%2B%20y%28b%29%7D%7B2%7D%20%20%5C%3A%20%29%20%5C%5C%20)
![o = ( \: \frac{ - 4 + 4}{2} \: , \: \frac{8 - 4}{2} ) \\](https://tex.z-dn.net/?f=o%20%3D%20%28%20%5C%3A%20%20%5Cfrac%7B%20-%204%20%2B%204%7D%7B2%7D%20%20%5C%3A%20%2C%20%5C%3A%20%20%5Cfrac%7B8%20-%204%7D%7B2%7D%20%29%20%5C%5C%20)
![o = ( \: \frac{0}{2} \: , \: \frac{4}{2} \: ) \\](https://tex.z-dn.net/?f=o%20%3D%20%28%20%5C%3A%20%20%5Cfrac%7B0%7D%7B2%7D%20%20%5C%3A%20%2C%20%5C%3A%20%20%5Cfrac%7B4%7D%7B2%7D%20%5C%3A%20%29%20%5C%5C%20)
![o = ( \: 0 \: , \: 2 \: )](https://tex.z-dn.net/?f=o%20%3D%20%28%20%5C%3A%200%20%5C%3A%20%2C%20%5C%3A%202%20%5C%3A%20%29)
This the coordinates of the centre.
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Equation of the circle has a form like this :
![( {x - m})^{2} + ({y - n})^{2} = {r}^{2}](https://tex.z-dn.net/?f=%28%20%7Bx%20-%20m%7D%29%5E%7B2%7D%20%20%2B%20%20%28%7By%20-%20n%7D%29%5E%7B2%7D%20%3D%20%20%7Br%7D%5E%7B2%7D%20%20)
In which :
![m = x - coordinate \: \: of \: \: the \: \: center \\](https://tex.z-dn.net/?f=m%20%3D%20x%20-%20coordinate%20%5C%3A%20%20%5C%3A%20of%20%5C%3A%20%5C%3A%20the%20%5C%3A%20%20%5C%3A%20center%20%20%20%5C%5C%20%20)
![n = y - coordinate \: \: of \: \: the \: \: center \\](https://tex.z-dn.net/?f=n%20%3D%20y%20-%20coordinate%20%5C%3A%20%20%5C%3A%20of%20%5C%3A%20%20%5C%3A%20the%20%5C%3A%20%20%5C%3A%20center%20%5C%5C%20)
![r = radius](https://tex.z-dn.net/?f=r%20%3D%20radius)
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![o = ( \: 0 \: , \: 2 \: )](https://tex.z-dn.net/?f=o%20%3D%20%28%20%5C%3A%200%20%5C%3A%20%2C%20%5C%3A%202%20%5C%3A%20%29)
![radius = 2 \sqrt{13}](https://tex.z-dn.net/?f=radius%20%3D%202%20%5Csqrt%7B13%7D%20)
Thus :
![({x - 0})^{2} + ({y - 2})^{2} = ({2 \sqrt{13} })^{2} \\](https://tex.z-dn.net/?f=%20%28%7Bx%20-%200%7D%29%5E%7B2%7D%20%2B%20%20%28%7By%20-%202%7D%29%5E%7B2%7D%20%20%3D%20%20%28%7B2%20%5Csqrt%7B13%7D%20%7D%29%5E%7B2%7D%20%20%20%5C%5C%20)
![{x}^{2} + ({y - 2})^{2} = 4 \times 13](https://tex.z-dn.net/?f=%20%7Bx%7D%5E%7B2%7D%20%20%2B%20%20%28%7By%20-%202%7D%29%5E%7B2%7D%20%20%3D%204%20%5Ctimes%2013)
![{x}^{2} + ({y - 2})^{2} = 52](https://tex.z-dn.net/?f=%20%7Bx%7D%5E%7B2%7D%20%20%2B%20%20%28%7By%20-%202%7D%29%5E%7B2%7D%20%3D%2052%20)
This is the equation of the circle .
And we're done....
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