Show that the equation z(z¯) + (a¯)z + a(z¯) + b = 0 with (a as element of complex number) and (b as element of real number) represents a circle in C.

Question

please help me, hints will work also as i don't know how to start this question :)

Answer 1

If z = x+yi and a = c+di then

zz¯ = (x+yi)(x-yi) = x^2+y^2
a¯z = (c-di)(x+yi) = (cx+dy)+(cy-dx)i
az¯ = (c+di)(x-yi) = (cx+dy)+(dx-cy)i

so, adding all that up you get

x^2+y^2 + 2cx+2dy + b = 0
(x+c)^2 + (y+d)^2 = c^2+d^2-b

Looks like a circle to me.

Answer 2

thanks... :)