with focal length $f$ and $a,b$ from the standard equation (up-down opening, origin centered): $$ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 $$
Using the definition that a hyperbola is the points whose distance from two points (focii) differ by a constant, the standard equation can be obtained, with $f^2-a^2$ in place of $b^2$ --- which rearranges to $a^2+b^2=f^2$.
It seems that $b$ is invented to make a nice standard form. Does it have geometric significance?
The $a^2+b^2=f^2$ equation itself has the form of the Pythagorean Theorem... but in a hyperbola $a,b,f$ do not measure the sides of a right triangle! Instead, $f$ is the distance from the center to each focii; $a$ is the distance from the center to the vertices; and $b$ does not appear to have a geometric representation. Far from forming a right triangle, the line segments measured by $a,f$ are colinear.
Is there an interpretation of $a$, $b$ and $f$ as a right triangle?
EDIT We can use the complex plane to get an intersection when $y=0$ (as for an ellipse), at $(ib,0)$. The distance between the x-axis and y-axis intersections (vertices) is $f$.
Though I don't see how that particular segment, of length $f$, has any geometric significance in itself.