Show that the chord of contact of tangents from the point $(a,-a)$ to the parabola $x^2=4ay$ has length $5a$

Question

Show that the chord of contact of tangents from the point $(a,-a)$ to the parabola $x^2=4ay$ has length $5a$. My current method uses the distance formula and takes super long, is there some easier method that i'm missing? Thanks in advance!

Answer 1

The first coordinates of the contact points are easily computed to $x_{1,2}=a\pm a\sqrt5$. Now the square of the distance of these points is clearly $$(x_2-x_1)^2+\left(\frac{1}{4a}(x_2^2-x_1^2)\right)^2.$$ Finally notice that $x_2-x_1=2\sqrt5$, $x_2+x_1=2a$ and that $x_2^2-x_1^2=(x_2-x_1)(x_2+x_1)=4\sqrt5a$.