I'm watching this video in Khan Academy math, and the goal is to find a relationship between a tangent line and a hyperbola. So given a hyperbola:
$$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $$
and a line $y = mx + c$,
Can we write down a relationship between the two equations? And Sal makes the claim that "the whole insight" is that these two lines only intersect at one point.
I don't understand this claim. I understand that if a line is a tangent line to a hyperbola, then it must intersect the hyperbola exactly once. But I don't think it holds that if a line intersects a hyperbola exactly once, then that line is a tangent line. For example, here is a line that intersects the hyperbola exactly once but is not tangent:
$$ y=\frac{xb}{a}+c $$
where $c > 0$. This line is parallel to one of the asymptotes of the hyperbola and therefore only intersects the hyperbola at one point. For example:
What am I missing here? Sal's whole line of reasoning seems predicated on the fact that if we solve for a line that intersects the hyperbola exactly once, we have a tangent line. That seems wrong to me.
This is similar to the situation with a parabola $y=ax^2$. The vertical lines intersect the parabola only once, but they're not tangent to it.
Where's the difference? Note that when we intersect with the line $x=h$, the equation becomes of degree $1$, not $2$. Tangency of a line to a conic is characterized by the degree $2$ equation having coincident roots, which algebraically is not the same as “unique solution”.
With an affine transformation, the hyperbola can be put in the simpler form $x^2-y^2=1$ and the transformation sends lines into lines and preserves intersections and tangency.
Consider a line $y=x+c$; then the intersections are computed with the equation $$ x^2-(x+c)^2=1 $$ that becomes $2cx=c^2-1$, which has a single solution when $c\ne0$ (the line would be an asymptote). But the equation has degree $1$, as you see.
Where is the “missing” intersection? It is an improper point! Projective geometry is necessary in order to fully understand what's going on.
The homogeneous equation of the hyperbola is $X^2-Y^2-Z^2=0$. The asymptotes have equations $X-Y=0$ and $X+Y=0$ respectively; if we intersect them with the hyperbola we obtain for the first $Z^2=0$, which is a degree $2$ equation with coincident roots. The point of tangency is $(1:1:0)$ in homogeneous coordinates. We get $(1:-1:0)$ for the other asymptote.
If we intersect the line $X-Y+cZ=0$ (that corresponds to $y=x+c$) we obtain the equation $$ X^2-(X+cZ)^2-Z^2=0 $$ which becomes $(1+c^2)Z^2+2cXZ=0$. If $c\ne0$, we get either $Z=0$ or $(1+c^2)Z+2cX=0$, which correspond to two points: $(1:1:0)$ and $$ \bigl((1+c^2):(1-c^2):-2c\bigr) $$ No tangency: the line has in common with the hyperbola a proper point (the latter) and the improper point of the asymptote $X-Y=0$.
Similarly for lines parallel to the other asymptote. In this context, asymptotes are simply lines that are tangent to the hyperbola at an improper point.