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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.
[tex]distance = diameter = d[/tex]
[tex]d = \sqrt{ ( \: {y(b) - y(a)} \: )^{2} + ( \: {x(b) - x(a)} \: )^{2} } \\ [/tex]
Now just need to put the coordinates :
[tex]d = \sqrt{ ({ - 4 - 8})^{2} + ( \: {4 - ( - 4)} \: )^{2} } \\ [/tex]
[tex]d = \sqrt{ ({ - 12})^{2} + ({4 + 4})^{2} } [/tex]
[tex]d = \sqrt{ ({ - 12})^{2} + ({8})^{2} } [/tex]
[tex]d = \sqrt{144 + 64} [/tex]
[tex]d = \sqrt{208} [/tex]
[tex]d = \sqrt{4 \times 52} [/tex]
[tex]d = \sqrt{4 \times 4 \times 13} [/tex]
[tex]d = \sqrt{ {4}^{2} \times 13 } [/tex]
[tex]d = 4 \sqrt{13} [/tex]
This is the diameter of the circle.
We know that :
[tex]diameter = 2 \times radius[/tex]
So :
[tex]4 \sqrt{13} = 2 \times radius[/tex]
Divide sides by 2
[tex] \frac{4 \sqrt{13} }{2} = \frac{2 \times radius}{2} \\ [/tex]
[tex]2 \sqrt{13} = radius[/tex]
Remember it.
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To find the equation of the circle except the radius , we need the coordinates of the centre.
The midpoint of the diameter is the centre of the circle .
So we have to find the midpoint of the diameter .
Look :
[tex]midpoint = center = o[/tex]
[tex]o = ( \: \frac{x(a) + x(b)}{2} \: , \: \frac{y(a) + y(b)}{2} \: ) \\ [/tex]
[tex]o = ( \: \frac{ - 4 + 4}{2} \: , \: \frac{8 - 4}{2} ) \\ [/tex]
[tex]o = ( \: \frac{0}{2} \: , \: \frac{4}{2} \: ) \\ [/tex]
[tex]o = ( \: 0 \: , \: 2 \: )[/tex]
This the coordinates of the centre.
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Equation of the circle has a form like this :
[tex]( {x - m})^{2} + ({y - n})^{2} = {r}^{2} [/tex]
In which :
[tex]m = x - coordinate \: \: of \: \: the \: \: center \\ [/tex]
[tex]n = y - coordinate \: \: of \: \: the \: \: center \\ [/tex]
[tex]r = radius[/tex]
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[tex]o = ( \: 0 \: , \: 2 \: )[/tex]
[tex]radius = 2 \sqrt{13} [/tex]
Thus :
[tex] ({x - 0})^{2} + ({y - 2})^{2} = ({2 \sqrt{13} })^{2} \\ [/tex]
[tex] {x}^{2} + ({y - 2})^{2} = 4 \times 13[/tex]
[tex] {x}^{2} + ({y - 2})^{2} = 52 [/tex]
This is the equation of the circle .
And we're done....
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