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## Homework Statement

Fairly straight forward question but I just can't see what's wrong.

A circle passes through the point (2,-4) and touches both the x-axis and the y-axis. Find the equations of the two circles which satisfy these conditions.

## Homework Equations

[tex] x^2 + y^2 + 2gx + 2fy + c = 0[/tex]

with a centre point [itex] c (-g,-f) [/itex]

[tex] r = \sqrt{g^2 + f^2 - c} [/tex]

Where r is the radius.

## The Attempt at a Solution

After drawing a diagram I concluded that r=g and r=f therefore g=f.

[tex] g = \sqrt{g^2 + f^2 - c} [/tex]

[tex] g^2 = g^2 + f^2 - c [/tex]

[tex] f^2 = c [/tex]

Since the point (2,-4) is on the circle it will satisfy :

[itex] x^2 + y^2 + 2gx + 2fy + c = 0[/itex]

[tex] (2)^2 + (-4)^2 + 2g(2) + 2f(-4) + c = 0 [/tex]

[tex] 4 + 16 + 4g -8f + c = 0 [/tex]

[tex] 20 + 4g -8f + c = 0 [/tex]

But [itex] f^2 = c [/itex] and [itex] f = g [/itex]

So

[tex] 20 + 4f - 8f + f^2 = 0[/tex]

[tex] 20 -4f + f^2 = 0 [/tex]

[tex] f^2 -4f +20 = 0 [/tex]

This quadratic only has complex roots.

Thanks for any help!