I would like to find the coefficients of a parabola passing through 2 known points. I also know the slope that the parabola should have at this 2 points. Let me show you a picture of the problem.
Known quantities: $(x_{N}, y_{N})$, $(x_{E}, y_{E})$, $\theta_{N}, \theta_{E}$.
Working with the equations I came up with: $$ \begin{align} x_{N} &= a \, y_{N}^{2} + b \, y_{N} + c \tag{1} \\ &\Rightarrow \frac{\partial x_{N}}{\partial y_{N}} = 2 a y_{N} + b = \frac{1}{\tan{\theta_{N}}} \tag{2} \\ x_{E} &= a \, y_{E}^{2} + b \, y_{E} + c \tag{3} \\ &\Rightarrow \frac{\partial x_{E}}{\partial y_{E}} = 2 a y_{E} + b = \frac{1}{\tan{\theta_{E}}} \tag{4} \\ \end{align} $$
I could write a system of three equations, for example:
$$ \left[ \begin{matrix} y_{N}^{2} & y_{N} & 1 \\ 2 y_{N} & 1 & 0 \\ y_{E}^{2} & y_{E} & 1 \end{matrix} \right] \left[ \begin{matrix} a \\ b \\ c \end{matrix} \right] = \left[ \begin{matrix} x_{N} \\ \frac{1}{\tan{\theta_{N}}} \\ x_{E} \end{matrix} \right] $$
but this will not consider the slope at point E.
If I write this system:
$$ \left[ \begin{matrix} y_{N}^{2} & y_{N} & 1 \\ 2 y_{N} & 1 & 0 \\ 2 y_{E} & 1 & 0 \end{matrix} \right] \left[ \begin{matrix} a \\ b \\ c \end{matrix} \right] = \left[ \begin{matrix} x_{N} \\ \frac{1}{\tan{\theta_{N}}} \\ \frac{1}{\tan{\theta_{E}}} \end{matrix} \right] $$
it will not consider the separation $x_{E} - x_{D}$.
How can I write a system that satisfy all my constraints?