Old Olympiad in Croatia 1971.

[rsoldo]

On which curve lies the set of centers of the circles whose segment on the $x$-axis is equal $2a$, and segment on the $y$-axis is equal $2b$?

[TOTAL]

$y^2+a^2=x^2+b^2$

[rsoldo]

Is it hyperbole?

[StaticZero]

Two sets of them, which are selected by the sign of $b^2 - a^2$.

Is it obvious from some trivial considerations that the set of centers has asymptotes?