Finding Equation of Parabola from given Focus and Base

1. Given a Parabola having Focus at a point \((x_f,y_f)\) and Equation Base as \(A_Bx + B_By + C_B=0\), it's Coordinates of Vertex \((x_v,y_v)\) can be found out using following steps
   1. Calculate the Signed Distance between the Focus and the Base as follows
      
      \(D_{FB}=\frac{A_Bx_f + B_By_f + C_B}{\sqrt{{A_B}^2 + {B_B}^2}}\)   ...(1)
   2. Calculate the Projection Point of Focus on the Base \((x_v,y_v)\) (which is the Vertex of the Parabola)as follows
      
      \(\begin{bmatrix}x_v\\y_v\end{bmatrix}=\begin{bmatrix}x_f\\y_f\end{bmatrix} - D_{FB}\begin{bmatrix}\frac{A_B}{\sqrt{{A_B}^2 + {B_B}^2}}\\\frac{B_B}{\sqrt{{A_B}^2 + {B_B}^2}}\end{bmatrix}\)   ...(2)
2. Now, since the Focus \((x_f,y_f)\) and the Vertex \((x_v,y_v)\) of the Parabola are known, the equation of the Parabola can be found out as given in Finding Equation of Parabola from given Focus and Vertex.