Question

Find the equation of the circle that has a diameter with endpoints (-4,8) and (4,-4)

Answer 1

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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`

`d = \sqrt{ ( \: {y(b) - y(a)} \: )^(2) + ( \: {x(b) - x(a)} \: )^(2) } \\`

Now just need to put the coordinates :

`d = \sqrt{ ({ - 4 - 8})^(2) + ( \: {4 - ( - 4)} \: )^(2) } \\`

`d = \sqrt{ ({ - 12})^(2) + ({4 + 4})^(2) }`

`d = \sqrt{ ({ - 12})^(2) + ({8})^(2) }`

`d = √(144 + 64)`

`d = √(208)`

`d = √(4 * 52)`

`d = √(4 * 4 * 13)`

`d = \sqrt{ {4}^(2) * 13 }`

`d = 4 √(13)`

This is the diameter of the circle.

We know that :

`diameter = 2 * radius`

So :

`4 √(13) = 2 * radius`

Divide sides by 2

`(4 √(13) )/(2) = (2 * radius)/(2) \\`

`2 √(13) = radius`

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 :

`midpoint = center = o`

`o = ( \: (x(a) + x(b))/(2) \: , \: (y(a) + y(b))/(2) \: ) \\`

`o = ( \: ( - 4 + 4)/(2) \: , \: (8 - 4)/(2) ) \\`

`o = ( \: (0)/(2) \: , \: (4)/(2) \: ) \\`

`o = ( \: 0 \: , \: 2 \: )`

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)`

In which :

`m = x - coordinate \: \: of \: \: the \: \: center \\`

`n = y - coordinate \: \: of \: \: the \: \: center \\`

`r = radius`

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`o = ( \: 0 \: , \: 2 \: )`

`radius = 2 √(13)`

Thus :

`({x - 0})^(2) + ({y - 2})^(2) = ({2 √(13) })^(2) \\`

`{x}^(2) + ({y - 2})^(2) = 4 * 13`

`{x}^(2) + ({y - 2})^(2) = 52`

This is the equation of the circle .

And we're done....

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