Solving Prop 86 Euclid's Data I was given the following problem:

Solve the equations of Prop-86 of the Data algebraically. Show that the two hyperbolas defined by the equations each have their axes as the asymptotes of each other.

All I can see about equation of Prop-86 is the system: $$xy=a\\\frac{y^2-b}{x^2}=\alpha$$
I don't understand what to do here. Am I supposed to solve this system of equations?
*Edit: This is the work I did. 
I understand all of it, until the end where I stated: "Therefore, two hyperbolas as defined have their axes as the asymptotes of the other." I wrote that, because based on the question I know this will be the conclusion, but I have no idea how that result actually came to be.

To me they appear to be two separate unconnected problems mentioned together, not in one system. First set/type is inclined to the axes and the second type/set is along the coordinate axis, is all the commonality that is there to it.
Plots of $$ \alpha x^2- y^2= \pm b, \;( \alpha= 3, b= \dfrac12) $$
show the desired hyperbolas pair.

Similarly in another statement of the same proposition  plots of
$$ xy= \pm b, \;(  b= \dfrac12) $$
show two different hyperbolas nested between (sharing) the same asymptote pair.

