We have

$$\overline z {z^3} + z{\overline z ^3} = 350$$

Substituting $$z = x + iy$$, we get

$$({x^2} + {y^2})({x^2} - {y^2}) = 175$$

$$({x^2} + {y^2})({x^2} - {y^2}) = 5 \times 5 \times 7$$

$${x^2} + {y^2} = 25$$

$${x^2} - {y^2} = 7$$

whose solutions are $$x = \pm 4$$ and $$y = \pm 3;x,y \in I$$.

Therefore, that is, area is found as $$8 \times 6 = 48$$ sq. unit.