Why are there only two tangents to a hyperbola from a point, instead of four?

The diagram shows a point $P$ between the arms of a hyperbola. It also shows how the rays emanating from $P$ fall into four regions (denoted "North", "South", "East", "West") bounded by $\overrightarrow{PA}$, $\overrightarrow{PB}$, $\overrightarrow{PC}$, $\overrightarrow{PD}$:

Rays in the North and South regions never hit the hyperbola at all. Rays in the East and West regions cut the hyperbola.

The boundary rays are special.

• $\overrightarrow{PA}$ and $\overrightarrow{PB}$ are tangent to the hyperbola (at "finite" points).

• $\overrightarrow{PC}$ and $\overrightarrow{PD}$ are parallel to the (invisible) asymptotes of the hyperbola. (It's often helpful to think of these as being tangent to the hyperbola at the "point at infinity", but that's not the sense in which you're using tangency.)

• (It's also good to know that the ray opposite $\overrightarrow{PC}$ separates the "Western" rays into two sub-regions: those that meet the hyperbola once, and those that meet it twice; likewise, the ray opposite $\overrightarrow{PD}$ sub-divides the "Eastern" rays.)

Moving $P$ can alter the nature of the boundaries a bit; for instance, if $P$ lies directly between the vertices (but, say, "west" of center), then $\overrightarrow{PA}$ and $\overrightarrow{PD}$ become the tangents, while $\overrightarrow{PB}$ and $\overrightarrow{PC}$ are asymptote-parallels. But the overall idea holds here, and apart from the case in which $P$ coincides with the hyperbola's center, there are always two tangents and two asymptote-parallels.

Anyway, the fact that a hyperbola has asymptotes (and thus region-bounding rays parallel to those asymptotes) distinguishes the geometry here from that of "two parabolas", which would have no asymptotes (and thus no corresponding region-bounding rays).