What does "$b$ - length of semi-conjugate axis" represent in the graph of hyperbola?

Question

In the standard equation of hyperbola, $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ where $b^2=a^2(e^2-1)$

If I were to draw the graph of hyperbola what would it represent in the graph? As $a$ represents the distance of vertex from the origin.

Answer 1

This is the length of the segment perpendicular to the major axis from vertex to either asymptote. Thus, as Blue notes, the asymptotes have slope $\pm\frac b a$.