Parabola

1. Any given General Quadratic Equation in 2 Variables mathematically represents a Planar Curve called Parabola if the Value of the Determinant of its E-Matrix is Non-Zero and Value of the Determinant of its e-Matrix is Zero.
2. A Parabola is A Set of All Points on a Plane which are Equidistant from Fixed Point and a Fixed Line. The Fixed Point is called the Focus of the Parabola and the Fixed Line is called the Directrix of the Parabola.
3. The Mid-Point between the Focus and the Directrix is called the Vertex of the Parabola.
4. The Line through the Vertex of Parabola Parallel to the Directrix of Parabola is called the Base of Parabola.
5. The Line through the Focus and Vertex of Parabola Perpendicular to the Directrix of Parabola is called the Axis of Symmetry of Parabola. The Axis of Symmetry Divides the Parabola into 2 Equal Halves.
6. The Distance Between Vertex and Focus of Parabola is called the Focal Length of the Parabola. The Focal Length is Half the Distance Between Focus and Directrix of Parabola.
7. The Line Segment Between 2 Points on Parabola Passing through its Focus, Parallel to its Directrix and Base and Perpendicular to its Axis of Symmetry is called the Latus Rectum of Parabola. The Length of Latus Rectum is 4 times the Focal Length of Parabola or Twice the Distance Between its Focus and Directrix.
8. The Ratio of the Distance Beween Focus and Vertext And Vertext and Directrix is called the Eccentricity. The Eccentricity of any Parabola is always 1.
9. In summary, following are the Parameters / Properties for any given Parabola
   1. Coordinates of Focus
   2. Coordinates of Vertex
   3. Coordinates of Points of Intersection of Parabola and the Latus Rectum
   4. Focal Length and Length of Latus Rectum
   5. Equation of the Directrix
   6. Equation of the Base
   7. Equation of the Latus Rectum
   8. Equation of the Axis of Symmetry
   9. Direction of the Axis of Symmetry
10. In 2 Dimensions, Parabolas can be of following 2 types
    1. Axis Aligned Parabolas: The Directrix of these Parabolas are Parallel to one of the Coordinate Axes. These can be of the following 4 subtypes
       1. Parabolas having Directix Parallel to \(X\)-Axis and opening in Upwards or Positive Direction of \(Y\)-Axis
       2. Parabolas having Directix Parallel to \(X\)-Axis and opening in Downwards or Negative Direction of \(Y\)-Axis
       3. Parabolas having Directix Parallel to \(Y\)-Axis and opening in Rightwards or Positive Direction of \(X\)-Axis
       4. Parabolas having Directix Parallel to \(Y\)-Axis and opening in Leftwards or Negative Direction of \(X\)-Axis
    2. Non-Axis Aligned Parabolas or Rotated Parabolas: The Directrix of these Parabolas are Not-Parallel to any Coordinate Axes.
11. In 2 Dimensions, Parabolas are represented using General Quadratic Equation in 2 Variables.
    
    For Axis Aligned Parabolas, the General Quadratic Equation in 2 Variables representing the Parabolas can be given in form of Standard Equations or Explicit Equations or Implicit Equations.
    
    For Non-Axis Aligned Parabolas or Rotated Parabolas, the General Quadratic Equation in 2 Variables representing the Parabolas can be given in form of Implicit Equations only.
12. Both in 2 and 3 Dimensions, Parabolas are also represented using Parametric Equations or Position Vector Expressions.