Finding Equation of Parabola from given Vertex and Latus Rectum

1. Given a Parabola having Vertex at a point \((x_v,y_v)\) and Equation Latus Rectum as \(A_Lx + B_Ly + C_L=0\), it's Coordinates of Focus \((x_f,y_f)\) can be found out using following steps
   1. Calculate the Signed Distance between the Vertex and Latus Rectum as follows
      
      \(D_{VL}=\frac{A_Lx_f + B_Ly_f + C_L}{\sqrt{{A_L}^2 + {B_L}^2}}\)   ...(1)
   2. Calculate the Projection Point of Vertex on the Latus Rectum \((x_f,y_f)\) (which is the Focus of the Parabola)as follows
      
      \(\begin{bmatrix}x_f\\y_f\end{bmatrix}=\begin{bmatrix}x_v\\y_v\end{bmatrix} - D_{VL}\begin{bmatrix}\frac{A_L}{\sqrt{{A_L}^2 + {B_L}^2}}\\\frac{B_L}{\sqrt{{A_L}^2 + {B_L}^2}}\end{bmatrix}\)   ...(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.