TITLE: Why only two tangents can be drawn to a parabola from a point outside it?
QUESTION [7 upvotes]: It's intuitively obvious to me that only two tangents can be drawn to a parabola from a point $(a,b)$ outside it but I want a mathematical pre-calculus proof of this.

REPLY [2 votes]: Say $\ell :ax+by+c=0$ is tangent to (given) parabola $y^2=2px$. We are seeking for $a,b,c$, not all $0$, given $p\ne 0$. We can assume $a=1$. Then the quadratic equation
$$y^2+2pby+2pc =0$$ must have only one solution on $y$, so $D=0$ and we get 
$$ 4p^2b^2-8pc =0$$ so $pb^2 = 2c\;\;\;(*)$. Since the point $T(x_0,y_0)$ is on tangent we have $c=-x_0-by_0$. Pluging in to $(*)$ we have:
$$ pb^2 = -2x_0-2by_0$$
So this equation is quadratic on $b$ so it has at most two solution and thus we have at most 2 tangents (no matter where $T$ lies).