In one of the problem book I am using, the following result is stated without explanation:
The straight line $ y= mx+ c$ is a secant, a tangent or passes outside the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2}=1$$
According to the condition
Secant: $c^2>a^2 m^2 - b^2$
Tangent: $c^2=a^2 m^2-b^2$
Outside: $c^2<a^2 m^2 - b^2$
What's the intuition behind the above result?
Basically you plug in $y=mx+c$ into the equation of the hyperbola, you obtain
$$ (b^2 - a^2m^2 )x^2 - 2mca^2 x - a^2 (b^2+c^2) = 0$$
so the discriminant is
$$ \Delta = 4a^2 (m^2c^2a^2 + (b^2 - a^2m^2)(b^2+c^2)) = 4a^2b^2 ( b^2+c^2-a^2m^2)$$
So it has two (resp. one, zero) intersection points if and only if $\Delta >0$ (resp. $=0$, $<0$).
The only thing has is less trivial is that a line is a tangent to the hyperbola if and only if they intersect at exactly one point. Note that this is not true for (e.g.) cubic curves: some tangent to a cubic curve intersect the curve at more than one points.