Solving Prop 86 Euclid's Data

Question

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.

Answer 1

This is not an answer to your question. What I am trying to do here is to give you an insight into it. That is why I am providing you with a diagram, which shows an example of the scenario you have described in your problem statement. If you are already aware of these facts, then you can ignore them and let me know that, so that I can delete my post.

You can visualize in the diagram that the axes and asymptotes of one hyperbola are the asymptotes and axes of the other hyperbola. The equations of the two hyperbolae are $$xy=8 \quad \mathrm{and}\quad \frac{y^2-5}{x^2}=1.$$

In other words, I have assumed $a=8$, $\alpha=1$ and $b=5$. In general, $a$ and $b$ can have any value, but $\alpha$ can have only one value, i.e. 1.

The two asymptotes of the hyperbola $xy=a$ are given by $x=0$ and $y=0$. Its axes are $y=\pm x$. They do not depend on the value of $a$. The two axes of the other hyperbola are $x=0$ and $y=0$, Therefore they too are independent of $\alpha$ and $b$. However, its asymptotes have the equations $y=\pm\sqrt{\alpha}x$, which depend on $\alpha$. That is why $\alpha$ cannot have a value other than 1.

I found an article at link, which, I think, describes a case of two hyperbolae and Prop 86. If you have access to this article, read it to see whether you can get more insight.

Answer 2

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.