Tangents, normals and focal chords

You will write functions for the tangent with a given slope, for the second point where a normal meets the parabola, and for the length of a focal chord. All three use $y^2 = 4ax$ and the parameter $t$ of the point $(at^2, 2at)$.

The idea. The tangent with slope $m$ is $y = mx + \dfrac{a}{m}$. The normal at $t$ meets the parabola again at $t' = -t - \dfrac{2}{t}$. The focal chord through $t$ has length $a\left(t + \dfrac{1}{t}\right)^2$.

Your tasks.

1. Finish tangent_intercept(a, m) to return the $y$-intercept of the tangent with slope m.
2. Finish normal_again(a, t) to return the point $(x, y)$ where the normal at t meets the parabola a second time.
3. Finish focal_chord(a, t) to return the length of the focal chord with one end at t.