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## Homework Statement

The circle x

^{2}+ y

^{2}- 4x - 4y + 4 = 0 is inscribed in a triangle, which has two of its sides along the coordinate axes. If the locus of the circumcentre is of the form

x + y - xy + k(x

^{2}+ y

^{2})

^{1/2}= 0. Find k.

## The Attempt at a Solution

The centre of the given circle is (2,2) and its radius is 2 units. Let the circumcentre be (h,k).

The intercepts of one side of the triangle on x and y axes are 2h and 2k respectively.

The distance between incentre and circumcentre is (R

^{2}- 2Rr)

^{1/2}, where R = (h

^{2}+ k

^{2})

^{1/2}

Square of it must be equal to (h -2)

^{2}+ (k -2)

^{2}

i.e. (R

^{2}- 2Rr) = (h -2)

^{2}+ (k -2)

^{2}

On solving I got

h + k - (h

^{2}+ k

^{2})

^{1/2}= 2

which is not of the form given.

Please point out my mistake if any.