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HallsofIvy $\displaystyle |z+1|= 2\sqrt{2}$ is a circle with center at (-1, 0) and radius $\displaystyle 2\sqrt{2}$ and $\displaystyle |z-5|= 2\sqrt{5}$ is a circle with center at (5, 0) and radius $\displaystyle 2\sqrt{5}$.
They are given by $\displaystyle (x+1)^2+ y^2= 8$ and $\displaystyle (x- 5)^2+ y^2= 20$. Subtracting one equation from the other gives a linear equation for x. After finding x, substitute back into either equation to get a quadratic equation for y. As Bruno J says, you want the point with positive y.