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# If $-3+ix^2y$ and $x^2+y+4i$ are conjugate of each other then $(x,y)$= ?

Let $z_1=-3+ix^2y$ and $z_2=x^2+y+4i$
$\Rightarrow\:\overline {z_1}=-3-ix^2y$ and $\overline {z_2}=x^2+y-4i$
Given that $\overline {z_1}=z_2$ and $\overline {z_2}=z_1$
$\Rightarrow\:-3-ix^2y=x^2+y+4i$ and $x^2+y-4i=-3+ix^2y$
$\Rightarrow\:x^2+y=-3\:\:and\:\:\:x^2y=-4$
$\Rightarrow\:\large\frac{-4}{y}$$+y=-3$
$\Rightarrow\:y^2+3y-4=0$
$\Rightarrow\:y=-4\:\:or\:\:y=1$
$\Rightarrow\:when\:y=-4,\:x=\pm1$
$and\:when\:y=1,\:x^2=-4$ which is not possible.
$\therefore \:(x,y)=(\pm1,-4)$
edited Jul 23, 2013