Given is an ellipse in the plane with half-axes a and b and a straight line $g$ (see image below). The line $g$ is of the form
\begin{align*} g(t)=\{x+tv: t \in \mathbb{R}\}. \end{align*},
where $x$ is the (first) intersection of the line with the ellipse.
I want to describe the line alternatively by choosing another base point, i. instead of $ x $ a new base point $ x ^ * $ should be chosen. A typical choice would be e.g. to take the closest point to the origin $ 0 $. Then you can describe $ x ^ * $ by
\begin{align*} x^*=x-\langle x,v \rangle v, \end{align*}
with the standard scalar product $\langle \cdot, \cdot \rangle$. My question ist now: How can i describe the "old" base point $x$ in terms of the new point $x^*$? It is clear, that $x=x^*+\langle x,v \rangle v$, but here i have still $x$ at the right side of the formula. I am hoping to find something like $x=x(x^*,v)$ or maybe depending on the half axes $a$ and $b$. Is this possible in a compact form?
HINT.
The equation of the ellipse can be written as $X^TAX=1$, where $A=\pmatrix{1/a^2 & 0 \\ 0 & 1/b^2\\}$.
Substitute there $X=x^*+vt$ and solve for $t$, retaining only the negative solution.