Tangents from a Point to a Parabola

How many tangents from an external point? From a general point $P(x_1,y_1)$ in the plane (not necessarily on the parabola), consider all lines through $P$ with the tangent form $y=mx+\dfrac{a}{m}$. Forcing this line through $P$:

This is a quadratic equation in $m$. A quadratic has (in general) two roots, so there are two values $m_1,m_2$ — the slopes of the two tangent lines that can be drawn from $P$ to the parabola. This confirms the general fact: from any point in the plane, exactly two tangents (real or complex) can be drawn to a parabola.

Perpendicular tangents and the directrix. Suppose the two tangents from $P$ happen to be mutually perpendicular, i.e. $m_1m_2=-1$. From the quadratic (1), the product of the roots is (constant term)/(leading coefficient):

Setting this equal to $-1$: $\dfrac{a}{x_1}=-1 \Rightarrow x_1=-a$ — which is exactly the equation of the directrix!

So: the locus of points from which the two tangents to a parabola are mutually perpendicular is precisely the directrix of the parabola. This elegant result is a shortcut used repeatedly: any "find $k$ so that perpendicular tangents can be drawn from $(p,q)$" problem reduces immediately to reading $p=-a$ off the given point, no quadratic required.

Worked Example 3 — verifying perpendicular tangents. Parabola $y^2=16x$ ($a=4$), tangents from $(-4,-9)$: substituting into (1): $-4m^2+9m+4=0$, i.e. $4m^2-9m+4=0$. Product of roots $m_1m_2=\dfrac{\text{constant}}{\text{coefficient of }m^2}=\dfrac44=1$... …