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Maths

The circle

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Circle basics

You must be able to recognise when an equation represents a circle.

The general equation of a circle normally appears in the form x^2 + y^2 + 2gx + 2fy + c = 0 where (-g, -f) is the centre of the circle and \sqrt {g^2 + f^2 - c} is the radius.

Notice that for the circle to exist g^2 + f^2 - c > 0.

Look at the following worked examples.

For x^2 + y^2 + 6x - 8y - 11 = 0

g^2 + f^2 - c = (3)^2 + (-4)^2 - ( - 11) = 36

so equation represents a circle with centre (-3, 4) and radius \sqrt {36} = 6

For x^2 + y^2 - 2x + 4y + 11 = 0

g^2 + f^2 - c = (-1)^2 + 2^2 - 11 = - 6

so x^2 + y^2 - 2x + 4y - 11 = 0 does not represent a circle.

For 3x^2 + 3y^2 - 6x + y - 9 = 0 we must write this starting x^2 + y^2 like this:

\eqalign{ x^2 + y^2 - 2x + {1 \over 3}y - 3 & = & 0\cr\cr g^2 + f^2 - c & = & ( - 1)^2 + ({1 \over 6})^2 - ( - 3) = 4{1 \over {36}} = {{145} \over {36}}\cr

so the equation represents a circle with centre (1, - {1 \over 6}) and radius \sqrt {{{145} \over {36}}} = {{\sqrt {145} } \over 6}

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