Condition for Tangency of a Hyperbola

The question: for which $m,c$ does $y=mx+c$ touch $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$?

Following exactly the same coefficient-comparison method as section 7.2.5 (line (1): $mx-y+c=0$; tangent-at-a-point (2): $\dfrac{x_1}{a^2}x-\dfrac{y_1}{b^2}y-1=0$), matching coefficients gives

Substituting into $\dfrac{x_1^2}{a^2}-\dfrac{y_1^2}{b^2}=1$:

So the condition of tangency is $\boxed{c^2=a^2m^2-b^2}$ — note the MINUS sign, the one place this differs from the ellipse's $c^2=a^2m^2+b^2$. The point of contact is

— again, note both coordinates carry a minus sign, unlike the ellipse's point of contact which has $+b^2/c$ for $y$. …