Lets say, a vertical hyperbola whose transverse axis is $y$ axis. Its parametric coordinates $(a\tan \theta, b\sec \theta)$. This $\theta$ represents the eccentric angle related with auxiliary circle. If any point of the hyperbola is located in the 1st quadrant (according to the proof), I understand the identity of $\theta$. But I can't find this angle in other quadrants. So, does this angle really have a geometric significance?
NB- I am a 11th grader. And don’t have much idea on Hyperbolic Trigonometry