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