Finding Equation of Ellipse from a given Focus, a Vertex and Eccentricity

1. Given Coordinates of Adjacent Focus and Vertex (\((x_{f1},y_{f1})\) and \((x_{v1},y_{v1})\) respectively) and Eccentricity \(e\) the following gives the steps for calculation of the Equation of the Ellipse
   1. Calculate the Distance \(d\) Between the given Focus and the Vertex as follows
      
      \(d=\sqrt{{(x_{f1}-x_{v1})}^2 + {(y_{f1}-y_{v1})}^2}\)   ...(1)
   2. Calculate the Length of Semi-Major Axis \(a\)
      
      We know that for Adjacent Vertex and Focus of Ellipse
      
      \(d=a-c\hspace{.5cm}\Rightarrow c=a-d\)   ...(2)
      
      where \(c\) is Half the Distance Between the 2 Foci of Ellipse or the Distance Between Center of the Ellipse and the given Focus. Also we know that
      
      \(e=\frac{c}{a}\hspace{.5cm}\Rightarrow c=ae\)   ...(3)
      
      From equations (2) and (3) above we have
      
      \(a-d=ae\hspace{.5cm}\Rightarrow a-ae=d\hspace{.5cm}\Rightarrow a(1-e)=d\hspace{.5cm}\Rightarrow a=\frac{d}{(1-e)}\)   ...(4)
   3. Calculate the Coordinates of the Center \((x_c,y_c)\) of Ellipse as follows
      
      \(\begin{bmatrix}x_c\\y_c\end{bmatrix}=\begin{bmatrix}x_{v1}\\y_{v1}\end{bmatrix} + a\begin{bmatrix}\frac{x_{f1}-x_{v1}}{d}\\\frac{y_{f1}-y_{v1}}{d}\end{bmatrix}\)   ...(5)
   4. Calculate the Coordinates of the other Focus \((x_{f2},y_{f2})\) of Ellipse as follows
      
      \(\begin{bmatrix}x_{f2}\\y_{f2}\end{bmatrix}=\begin{bmatrix}x_c\\y_c\end{bmatrix} + c\begin{bmatrix}\frac{x_{f1}-x_{v1}}{d}\\\frac{y_{f1}-y_{v1}}{d}\end{bmatrix}=\begin{bmatrix}x_c\\y_c\end{bmatrix} + ae\begin{bmatrix}\frac{x_{f1}-x_{v1}}{d}\\\frac{y_{f1}-y_{v1}}{d}\end{bmatrix}\)   ...(6)
   5. Once we get Coordinates of Both the Foci and the Length of the Semi Major Axis the Equation of the Ellipse can be determined as given in the topic Finding Equation of Ellipse from given 2 Foci and Major Axis Length.
2. Given Coordinates of Non-Adjacent Focus and Vertex (\((x_{f2},y_{f2})\) and \((x_{v1},y_{v1})\) respectively) and Eccentricity \(e\) the following gives the steps for calculation of the Equation of the Ellipse
   1. Calculate the Distance \(d\) Between the given Focus and the Vertex as follows
      
      \(d=\sqrt{{(x_{f2}-x_{v1})}^2 + {(y_{f2}-y_{v1})}^2}\)   ...(7)
   2. Calculate the Length of Semi-Major Axis \(a\)
      
      We know that for Non-Adjacent Vertex and Focus of Ellipse
      
      \(d=a+c\hspace{.5cm}\Rightarrow c=d-a\)   ...(8)
      
      where \(c\) is Half the Distance Between the 2 Foci of Ellipse or the Distance Between Center of the Ellipse and the given Focus. Also we know that
      
      \(e=\frac{c}{a}\hspace{.5cm}\Rightarrow c=ae\)   ...(9)
      
      From equations (8) and (9) above we have
      
      \(d-a=ae\hspace{.5cm}\Rightarrow a+ae=d\hspace{.5cm}\Rightarrow a(1+e)=d\hspace{.5cm}\Rightarrow a=\frac{d}{(1+e)}\)   ...(10)
   3. Calculate the Coordinates of the Center \((x_c,y_c)\) of Ellipse as follows
      
      \(\begin{bmatrix}x_c\\y_c\end{bmatrix}=\begin{bmatrix}x_{v1}\\y_{v1}\end{bmatrix} + a\begin{bmatrix}\frac{x_{f2}-x_{v1}}{d}\\\frac{y_{f2}-y_{v1}}{d}\end{bmatrix}\)   ...(11)
   4. Calculate the Coordinates of the other Focus \((x_{f1},y_{f1})\) of Ellipse as follows
      
      \(\begin{bmatrix}x_{f1}\\y_{f1}\end{bmatrix}=\begin{bmatrix}x_c\\y_c\end{bmatrix} - c\begin{bmatrix}\frac{x_{f2}-x_{v1}}{d}\\\frac{y_{f2}-y_{v1}}{d}\end{bmatrix}=\begin{bmatrix}x_c\\y_c\end{bmatrix} - ae\begin{bmatrix}\frac{x_{f2}-x_{v1}}{d}\\\frac{y_{f2}-y_{v1}}{d}\end{bmatrix}\)   ...(12)
   5. Once we get Coordinates of Both the Foci and the Length of the Semi Major Axis the Equation of the Ellipse can be determined as given in the topic Finding Equation of Ellipse from given 2 Foci and Major Axis Length.