What is the reason behind the Pythagorean relation in a hyperbola?

Solution 1:

Consider the following image:

Here, I have a drawn a hyperbola of the form $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,$$ for some $0 < a < b$. The asymptotes $y/b = \pm x/a$ have also been shown, and it is easy to see algebraically that the asymptotes indeed must be these equations.

The green rectangle is drawn such that the horizontal width is the distance $2a$, thus opposite sides are tangent to the vertices of the hyperbola at $(\pm a, 0)$. The vertical height of the rectangle is chosen such that the rectangle's vertices are on the asymptotes, thus the height is $2b$ and the rectangle's vertices are $(\pm a, \pm b)$ where the signs can be chosen independently.

The green circle is simply the circumscribing circle to the rectangle. The intersection of this circle with the $x$-axis is the location of the hyperbola's foci, at $(\pm c, 0)$. It is natural to see, then, that $$c^2 = a^2 + b^2,$$ from our description of the rectangle's vertices and the Pythagorean relationship that relates the rectangle's width and height to the circumradius.

The question, then, is why the focus happens to be located at the point of intersection of this circle with the coordinate axis? The reason is that the definition of the hyperbola is the locus of all points whose absolute difference of distances from the foci is constant. So for such a choice of $c$, it is fairly easy (though tedious) to show that points $(x,y)$ satisfying the above equation for the hyperbola will have an absolute difference of distances from $(\pm c,0)$ a constant. What is this constant value?