Scattering from a hard ellipsoid

[Present]

Homework Statement

a,b is constant and s is impact parameter, θ is scattering angle.
i know that ψ in the picture is <ψ=(π-θ)/2>

Homework Equations

differential scattering crossection dσ/dΩ = (s/sinΘ) I ds/dθ I

and σ(θ)=∫(dσ/dΩ)dΩ

The Attempt at a Solution

i guessed, first step is that finding a relation of impact parameter s and scattering angle θ.
but i couldn't. This picture is my attempt at a solution.

i think for getting a relation of <s and θ> to use tan(θ/2) but i do not find.
How can i get impact parameter s(θ)?

 

[haruspex]

calculate, or look up, the equation for the tangent to an ellipse.