Find the real values of the parameter a for which at least one complex number z=x+iy satisfies $\displaystyle |z+\sqrt{2}|=a^{2}-3a+2$ and $\displaystyle |z+i\sqrt{2}|<a^2$
2. |z- a|= r can be interpreted as a circle, in the complex plane, with center at a and radius r. In particular, $\displaystyle |z+ i\sqrt{2}|= a^2$ is a circle with center at $\displaystyle i\sqrt{2}$ and radius $\displaystyle a^2$. The set $\displaystyle |z+ i\sqrt{2}|< a^2$ is the interior of that circle. Similarly, $\displaystyle |z+ \sqrt{2}|= a^2- 3a+ 2$ is a circle with center at $\displaystyle \sqrt{2}$ and radius $\displaystyle a^2- 3z+ 2$. There will be at least one point on that circle, in the interior of the first circle if and only if those circles over lap. That is true if and only if the sum of their radii, $\displaystyle 2a^2- 3a+ 2$, is greater than the distance between the centers, $\displaystyle |\sqrt{2}- i\sqrt{2}|= \sqrt{2}|1- i|= 2$.
That is, there will be at least one such complex number as long as $\displaystyle 3a^2- 3a+ 2\ge 2$.