For what values of $x$ and $y$, the numbers $-3 + i x^2 y$ and $x^2 + y + 4i$ are complex conjugate of each other?
Let the given numbers be
$z_1 = -3 + i x^2 y$ $\;$ and $\;$ $z_2 = x^2 + y + 4i$
Complex conjugate of $z_1 = \overline{z_1} = -3 - i x^2 y$
Given: $\;$ $\overline{z_1} = z_2$
i.e. $\;$ $-3 - i x^2 y = x^2 + y + 4i$ $\;\;\; \cdots \; (1)$
Equating the real parts on either side of equation $(1)$ we have,
$-3 = x^2 + y$ $\;\;\; \cdots \; (2a)$
Equating the imaginary parts on either side of equation $(1)$ we have,
$- x^2 y = 4$ $\;\;\; \cdots \; (2b)$
From equation $(2a)$, $\;$ $x^2 = -3 - y$ $\;\;\; \cdots \; (3)$
Substituting the value of $x^2$ in equation $(2b)$ we have,
$\left(3 + y\right) y = 4$
i.e. $\;$ $y^2 + 3y-4 = 0$
i.e. $\;$ $\left(y + 4\right) \left(y - 1\right) = 0$
$\implies$ $y = -4$ $\;$ or $\;$ $y = 1$
Substituting $y = -4$ in equation $(3)$ we have,
$x^2 = 1$ $\implies$ $x = \pm 1$
Substituting $y = 1$ in equation $(3)$ we have,
$x^2 = -4$ $\implies$ $x = \pm 2 i$