Intersection between a line and an ellipse

Lets consider a line having slope $\zeta$ and containing the test point $A$.

Figure 2: line/ellipse intersection

For well chosen $\zeta$ values, this line intersects the ellipse. Let $(r',z')$ be the cartesian coordinates of the intersection point $P(\varphi)$:

$$\left\{\begin{aligned}r' &= r-k\cos\zeta\\ z' &= z-k\sin\zeta \end{aligned}\right.$$

where $k$ is the signed distance between the test point $A$ and the intersection point $P(\varphi)$. $k$ is positive if the test point is outside of the ellipse and negative otherwise. Note that the sign of $k$ implies the choice of the line orientation, so depending on the test point location inside or outside of the ellipse, we have to choose either $\zeta$ or $\pi-\zeta$; the following equations take care of this choice. Since the intersection point $P(\varphi)$ belongs to the ellipse, its coordinates verify equation:

$$\left(\frac{r'}{a_e}\right)^2+\left(\frac{z'}{(1-f)a_e}\right)^2 = 1$$

introducing the test point coordinates and the signed distance $k$

$$\left(\frac{r-k\cos\zeta}{a_e}\right)^2+\left(\frac{z-k\sin\zeta}{(1-f)a_e}\right)^2 = 1$$

$$\Leftrightarrow\quad (1-f)^2[(r-k\cos\zeta)^2-a_e^2] + (z-k\sin\zeta)^2 = 0$$

$$\Leftrightarrow\quad a k^2 - 2b k + c = 0 \quad \left\{\begin{aligned}a &= (1-f)^2\cos^2\zeta+\sin^2\zeta\\... ...(1-f)^2r\cos\zeta+z\sin\zeta\\ c &= (1-f)^2(r^2-a_e^2)+z^2 \end{aligned}\right.$$ (5)