Complex-city

Let $z_1=x+iy$ and $z_2=p+iq$ ,

$\therefore$ $x^{3}+3x^{2}iy+3xi^{2}y^{2}+i^{3}y^{3}=x-iy$

$\Rightarrow$ $(x^{3}-3xy^{2})+i(3x^{2}y-y^{3})=x-iy$

Equating real and imaginery parts, we have $x^{3}-3xy^{2}=x$ and $3x^{2}y-y^{3}=-y$

Thus we have to solve the following $4$ pairs of simultaneous equations:

$x=0$ , $y=0$ ; $x=0$ , $3x^{2}-y^{2}+1=0$ ; $y=0$ , $x^{2}-3y^{2}-1=0$ ; $x^{2}-3y^{2}-1=0$ , $3x^{2}-y^{2}+1=0$

Solve to get $z_1=0,\pm1,\pm i$

$arg(\frac{p^{2}+q^{2}-4+4qi}{(p+2)^{2}+q^{2}})=\frac{\pi}{4}$

$\Rightarrow$ $\frac{4q}{p^{2}+q^{2}-4}=tan\frac{\pi}{4}$

$\Rightarrow$ $p^{2}+(q-2)^{2}=8$

$\therefore$ $|z_2-2i|=\sqrt{8}$

$\Rightarrow$ $a=5, b=2, c=8$

$Ans: 15$ $\ddot\smile$