I have a parabola defined in terms $3$ points i.e $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, consider $(x_1, y_1)$ be start point, $(x_2,y_2)$ be the vertex and $(x_3, y_3)$ be the end point and also provided starting angle and ending angle. Also, I have line segment defined in terms of two points $(m_1, n_1)$ and $(m_2, n_2)$.
I want to get the final equation that provides the intersection points of the line segment and a parabola.
consider this reference image:
Since it passes through $(x_1,y_1)$, parabola is of form $$f(x)=a(x-x_1)(x-b)+y_1$$ Then plug $(x_2,y_2),(x_3,y_3)$ in order to get values of $a$ and $b$. Then create equation of line using two point from and finally intersect both quations