Finding Parameters of Axis Aligned Ellipse from Standard Coordinate Equation

1. As given in Derivation of Standard and Implicit Coordinate Equation for Axis Aligned Ellipses, the Standard Coordinate Equation for Axis Aligned Ellipses are given as
   
   \(\frac{x-x_c}{a^2} + \frac{y-y_c}{b^2} = \pm\hspace{1mm}1\)   (For Real and Imaginary Ellipses having Major Axes Parallel to \(X\)-Axis)...(1)
   
   \(\frac{y-y_c}{a^2} + \frac{x-x_c}{b^2} = \pm\hspace{1mm}1\)   (For Real and Imaginary Ellipses having Major Axes Parallel to \(Y\)-Axis)...(2)
2. The following gives the Formulae for finding out Various Parameters of Ellipse given the Standard Coordinate Equation
   1. Coordinates of the Center: \((x_c,y_c)\)
   2. Length of Semi-Major Axis: \(a\)
   3. Length of Major Axis: \(L=2a\)
   4. Length of Semi-Minor Axis: \(b\)
   5. Length of Minor 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 Major 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 Major Axis: \(y=y_c\)
      Equation of Minor Axis: \(x=x_c\)
      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 Ellipse: \((x_c+c,y_c+\frac{l^2}{2L})\),\((x_c+c,y_c-\frac{l^2}{2L})\)   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 Ellipse: \((x_c-c,y_c+\frac{l^2}{2L})\),\((x_c-c,y_c-\frac{l^2}{2L})\)   OR   \((x_c-c,y_c+\frac{b^2}{a})\),\((x_c-c,y_c-\frac{b^2}{a})\)
   10. If Major 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 Major Axis: \(x=x_c\)
       Equation of Minor Axis: \(y=y_c\)
       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 Ellipse: \((x_c+\frac{l^2}{2L},y_c+c)\),\((x_c-\frac{l^2}{2L},y_c+c)\)   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 Ellipse: \((x_c+\frac{l^2}{2L},y_c-c)\),\((x_c-\frac{l^2}{2L},y_c-c)\)   OR   \((x_c+\frac{b^2}{a},y_c-c)\),\((x_c-\frac{b^2}{a},y_c-c)\)