Finding Parameters of Axis Aligned Hyperbola from Implicit Coordinate Equation

1. As given in Derivation of Standard and Implicit Coordinate Equation for Axis Aligned Hyperbolas, the Implicit Coordinate Equation for Axis Aligned Hyperbolas are given as
   
   \(Ax^2 + Cy^2 + Dx + Ey +F=0\)   ...(1)
   
   \(\Rightarrow Ax^2 + Dx + Cy^2 + Ey +F=0\)
   
   \(\Rightarrow A(x^2 + \frac{D}{A}x) + C(y^2 + \frac{E}{C}y) +F=0\)   ...(2)
   
   Completing the Squares in equation (2) we get
   
   \(A(x^2 + 2\frac{D}{2A}x + \frac{D^2}{4A^2} - \frac{D^2}{4A^2}) + C(y^2 + 2\frac{E}{2C}y + \frac{E^2}{4C^2} - \frac{E^2}{4C^2}) + F =0\)
   
   \(\Rightarrow A(x^2 + 2\frac{D}{2A}x + \frac{D^2}{4A^2}) + C(y^2 + 2\frac{E}{2C}y + \frac{E^2}{4C^2}) + F - \frac{D^2}{4A} - \frac{E^2}{4C} =0\)
   
   \(\Rightarrow A{(x + \frac{D}{2A})}^2 + C{(y + \frac{E}{2C})}^2 + F - \frac{D^2}{4A} - \frac{E^2}{4C}=0\)
   
   \(\Rightarrow A{(x + \frac{D}{2A})}^2 + C{(y + \frac{E}{2C})}^2 = \frac{D^2}{4A} + \frac{E^2}{4C} - F\)   ...(3)
   
   Setting \(\frac{D^2}{4A} + \frac{E^2}{4C} - F = \mathbf{G}\) in equation (3) we get
   
   \(A{(x + \frac{D}{2A})}^2 + C{(y + \frac{E}{2C})}^2 = G\)
   
   \(\Rightarrow A{\Large \frac{{(x + \frac{D}{2A})}^2}{G}} + C{\Large \frac{{(y + \frac{E}{2C})}^2}{G}} = 1\)
   
   \({\Rightarrow \Large\frac{{(x + \frac{D}{2A})}^2}{\frac{G}{A}}} + {\Large \frac{{(y + \frac{E}{2C})}^2}{\frac{G}{C}}} = 1\)   ...(4)
   
   If equation (1) represents an Implicit Equation of Hyperbola, values of \({\Large \frac{G}{A}}\) and \({\Large \frac{G}{C}}\) given in equation (4) will have Different Signs.
   
   If the values of \({\Large \frac{G}{A}} > 0\) and \({\Large \frac{G}{C}} < 0\), then the equation (1) represents a \(X\)-Transverse Hyperbola and equation (4) gives it's Stardard Coordinate Equation.
   
   If the values of \({\Large \frac{G}{A}} < 0\) and \({\Large \frac{G}{C}} > 0\), then the equation (1) represents a \(Y\)-Transverse Hyperbola and equation (4) gives it's Stardard Coordinate Equation.
2. Using equation (4) Various Parameters of Hyperbola given by Implicit Equation (1) can be given as
   1. Coordinates of the Center: \((x_c,y_c)=(-\frac{D}{2A},-\frac{E}{2C})\)
   2. Length of Semi-Transverse Axis: \(a=if(\frac{G}{A}>0, \sqrt{|\frac{G}{A}|},\sqrt{|\frac{G}{C}|})\)
   3. Length of Transverse Axis: \(L=2a\)
   4. Length of Semi-Conjugate Axis: \(b=if(\frac{G}{A}<0, \sqrt{|\frac{G}{A}|},\sqrt{|\frac{G}{B}|})\)
   5. Length of Conjugate Axis: \(l=2b\)
   6. Distance Between 2 Foci: \(F=\sqrt{L^2+l^2}=2c=2\sqrt{a^2+b^2}\)
   7. Eccentricity \(e\) : \(\frac{F}{L}=\frac{c}{a}\)
   8. Length for Latus Rectum : \(\frac{l^2}{L}=\frac{2b^2}{a}\)
   9. If Transverse Axis is Parallel to \(X\)-Axis then
      Angular Orientation: 0°
      Coordinates of the 2 Foci: \((x_c + c,y_c),\hspace{.3cm}(x_c - c,y_c)\)
      Coordinates of the 2 Vertices: \((x_c + a,y_c),\hspace{.3cm}(x_c - a,y_c)\)
      Coordinates of the 2 Co-Vertices: \((x_c,y_c+b),\hspace{.3cm}(x_c,y_c-b)\)
      Equation of Transverse Axis: \(y=y_c\)
      Equation of Conjugate Axis: \(x=x_c\)
      Equation of Conjugate Hyperbola: \(\frac{{(y-y_c)}^2}{b^2} - \frac{{(x-x_c)}^2}{a^2} = 1\)
      Equation of 2 Directrices: \(x=x_c+\frac{L^2}{2F},\hspace{.3cm} x=x_c-\frac{L^2}{2F}\)   OR   \(x=x_c+\frac{a^2}{c},\hspace{.3cm} x=x_c-\frac{a^2}{c}\)
      Equation of 2 Latus Recta: \(x=x_c+c,\hspace{.5cm} x=x_c-c\)
      Coordinates of Points of Intersection of Latus Rectum-1 and Hyperbola: \((x_c+c,y_c+\frac{l^2}{2L})\),\((x_c+c,y_c-\frac{l^2}{2L})\)   OR   \((x_c+c,y_c+\frac{b^2}{a})\),\((x_c+c,y_c-\frac{b^2}{a})\)
      Coordinates of Points of Intersection of Latus Rectum-2 and Hyperbola: \((x_c-c,y_c+\frac{l^2}{2L})\),\((x_c-c,y_c-\frac{l^2}{2L})\)   OR   \((x_c-c,y_c+\frac{b^2}{a})\),\((x_c-c,y_c-\frac{b^2}{a})\)
   10. If Transverse Axis is Parallel to \(Y\)-Axis then
       Angular Orientation: 90°
       Coordinates of the 2 Foci: \((x_c,y_c + c),\hspace{.3cm}(x_c,y_c - c)\)
       Coordinates of the 2 Vertices: \((x_c,y_c + a),\hspace{.3cm}(x_c,y_c - a)\)
       Coordinates of the 2 Co-Vertices: \((x_c+b,y_c),\hspace{.3cm}(x_c-b,y_c)\)
       Equation of Transverse Axis: \(x=x_c\)
       Equation of Conjugate Axis: \(y=y_c\)
       Equation of Conjugate Hyperbola: \(\frac{{(x-x_c)}^2}{b^2} - \frac{{(y-y_c)}^2}{a^2} = 1\)
       Equation of 2 Directrices: \(y=y_c+\frac{L^2}{2F},\hspace{.3cm} y=y_c-\frac{L^2}{2F}\)   OR   \(y=y_c+\frac{a^2}{c},\hspace{.3cm} y=y_c-\frac{a^2}{c}\)
       Equation of 2 Latus Recta: \(y=y_c+c,\hspace{.5cm} y=y_c-c\)
       Coordinates of Points of Intersection of Latus Rectum-1 and Hyperbola: \((x_c+\frac{l^2}{2L},y_c+c)\),\((x_c-\frac{l^2}{2L},y_c+c)\)   OR   \((x_c+\frac{b^2}{a},y_c+c)\),\((x_c-\frac{b^2}{a},y_c+c)\)
       Coordinates of Points of Intersection of Latus Rectum-2 and Hyperbola: \((x_c+\frac{l^2}{2L},y_c-c)\),\((x_c-\frac{l^2}{2L},y_c-c)\)   OR   \((x_c+\frac{b^2}{a},y_c-c)\),\((x_c-\frac{b^2}{a},y_c-c)\)