Explain why none of the lines through a point inside a hyperbola is a tangent line to that hyperbola?

Question

Explain why none of the lines through a point inside a hyperbola is a tangent line to that hyperbola?

I'm thinking since points must be on the hyperbola in order to be tangent, then they can't be since they're inside the hyperbola.

Answer 1

A hyperbola is given by a quadratic equation in $x$ and $y$, so assuming that a line $ax+by=c$ goes through a point on the hyperbola and it is also a tangent line to the hyperbola, there is a quadratic equation in $x$ (we get that by eliminating $y$) with three roots accounted with multiplicity. That obviously cannot happen, since $\mathbb{R}[x]$ is an integral domain: any polynomial with degree $d$ has at most $d$ real roots accounted with multiplicity.

Everything changes dramatically if we consider an algebraic curve like $$ y^2 = x^3+ax+b $$ over $\mathbb{C}$: in such a case every tangent line has to meet the curve at another point. In facts, that gives a group structure for an elliptic curve and a duplication formula.