Finding Parameters of Axis Aligned Hyperbola from Standard Coordinate Equation

As given in Derivation of Standard and Implicit Coordinate Equation for Axis Aligned Hyperbolas, the Standard Coordinate Equation for Axis Aligned Hyperbolas is given as is given as

\(\frac{x-x_c}{a^2} - \frac{y-y_c}{b^2} = 1\) (For Hyperbolas having Transverse Axes Parallel to \(X\)-Axis)...(1)

\(\frac{y-y_c}{a^2} - \frac{x-x_c}{b^2} = 1\) (For Hyperbolas having Transverse Axes Parallel to \(Y\)-Axis)...(2)
The following gives the Formulae for finding out Various Parameters of Hyperbola given the Standard Coordinate Equation
1. Coordinates of the Center: \((x_c,y_c)\)
2. Length of Semi-Transverse Axis: \(a\)
3. Length of Transverse Axis: \(L=2a\)
4. Length of Semi-Conjugate Axis: \(b\)
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. Equation the 2 Asymptotic Lines: \(a(y-y_c)=b(x-x_c),\hspace{.5cm}a(y-y_c)=-b(x-x_c)\)
10. If Transverse Axis is Parallel to \(X\)-Axis then
  Angular Orientation: 0°
  Coordinate of the 2 Foci: \((x_c + c,y_c),\hspace{.3cm}(x_c - c,y_c)\)
  Coordinate of the 2 Vertices: \((x_c + a,y_c),\hspace{.3cm}(x_c - a,y_c)\)
  Coordinate 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   \((x_c-c,y_c+\frac{b^2}{a})\),\((x_c-c,y_c-\frac{b^2}{a})\)
11. If Transverse Axis is Parallel to \(Y\)-Axis then
  Angular Orientation: 90°
  Coordinate of the 2 Foci: \((x_c,y_c + c),\hspace{.3cm}(x_c,y_c - c)\)
  Coordinate of the 2 Vertices: \((x_c,y_c + a),\hspace{.3cm}(x_c,y_c - a)\)
  Coordinate 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   \((x_c+\frac{b^2}{a},y_c-c)\),\((x_c-\frac{b^2}{a},y_c-c)\)

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