in hyperbolas, is it possible for the hyperbola to cross the asymptote and then get closer to it?
No. An asymptote is never reached/touched, that is the whole idea of an asymptote - it is a limit which is approached and as you get to infinity the 'gap' between the approaching line and the asymptote tends to zero.
If the line in your problem is crossing an asymptote,you have made an error somewhere.
You might be getting confused with functions like the tangent function where it approaches the same asymptote from the positive and negative direction - but still never touches. Have a look at the graph of tan (x) to see this.