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Question

If z=x+iy, then show that z¯¯¯z+2(z+¯¯¯z)+a=0, where aR, represents a circle.

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Solution

z=x+iy¯¯¯z=xiy
Therefore,
z¯¯¯z+2(z+¯¯¯z)+a=0
(x+iy)(xiy)+2(x+iy+xiy)+a=0
x2+y2+4x+a=0
(x+2)2+(y0)2=(4a)2, which represents a circle for all a4.

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