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General Quadratic Equations in 2 Variables and Conic Sections

1. Any equation given in the forms of the following are General Quadratic Equations in 2 Variables
   
   \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F =\begin{bmatrix}x & y & 1\end{bmatrix} \begin{bmatrix} 2A & B & D \\B & 2C & E \\ D & E & 2F\end{bmatrix}\begin{bmatrix}x \\ y \\ 1\end{bmatrix}=\begin{bmatrix}x & y & 1\end{bmatrix}\begin{bmatrix} A & \frac{1}{2}B & \frac{1}{2}D \\\frac{1}{2}B & C & \frac{1}{2}E \\ \frac{1}{2}D & \frac{1}{2}E & F\end{bmatrix}\begin{bmatrix}x \\ y \\ 1\end{bmatrix}=0\)      ...(1)
   
   \(Ax^2 + 2Bxy + Cy^2 + 2Dx + 2Ey + F =\begin{bmatrix}x & y & 1\end{bmatrix} \begin{bmatrix} A & B & D \\B & C & E \\ D & E & F\end{bmatrix}\begin{bmatrix}x \\ y \\ 1\end{bmatrix}=0\)      ...(2)
   
   Please note that in such equations Atleast 1 of Co-efficients \(A\) (i.e. Co-efficient of \(x^2\)), \(B\) (i.e. Co-efficient of \(xy\)) or \(C\) (i.e. Co-efficient of \(y^2\)) Must be Non-Zero.
   
   Also note that both equation (1) and equation (2) are just different representation of the same equation.
2. The General Quadratic Equations in 2 Variables geometrically represent a Set of Planar Curves known as Conic Sections or Conics. These Curves are formed on a Plane when the Plane Intersects a 3-Dimensional Cone in different Orientations.
3. Determining the Type of Curve and its Related Parameters/Properties that any given General Quadratic Equations in 2 Variables represents is called Conic Section Analysis.
4. For analysing any given General Quadratic Equations in 2 Variables , a certain set of Matrices, Determinants and Related Algebraic Entities are used. Representation of these Matrices, Determinants and Related Algebraic Entities differ depending on whether Any Given Quadratic Equation is Analysed in form of equation (1) or equation (2).
   
   Following are the Matrices, Determinants and Related Algebraic Entities used to analyse the General Quadratic Equations in 2 Variables when they are Analysed in form of equation (1)
   
   \(e=\begin{bmatrix} 2A & B \\B & 2C \end{bmatrix} \hspace{.3cm} OR \hspace{.3cm} e=\begin{bmatrix} A & \frac{1}{2}B \\\frac{1}{2}B & C \end{bmatrix}\)
   
   \(J=|e|=\begin{vmatrix} 2A & B \\B & 2C \end{vmatrix}= 4AC - B^2 \hspace{.3cm} OR \hspace{.3cm} J=|e|=\begin{vmatrix} A & \frac{1}{2}B \\\frac{1}{2}B & C \end{vmatrix}= AC - \frac{1}{4}B^2\)
   
   \(E=\begin{bmatrix} 2A & B & D \\B & 2C & E \\ D & E & 2F\end{bmatrix} \hspace{.3cm} OR \hspace{.3cm} E=\begin{bmatrix} A & \frac{1}{2}B & \frac{1}{2}D \\\frac{1}{2}B & C & \frac{1}{2}E \\ \frac{1}{2}D & \frac{1}{2}E & F\end{bmatrix}\)
   
   \(\Delta=|E|=\begin{vmatrix} 2A & B & D \\B & 2C & E \\ D & E & 2F\end{vmatrix}=8ACF + 2BDE - 2AE^2 - 2CD^2 - 2FB^2\)
   OR
   \(\Delta=|E|=\begin{vmatrix} A & \frac{1}{2}B & \frac{1}{2}D \\\frac{1}{2}B & C & \frac{1}{2}E \\ \frac{1}{2}D & \frac{1}{2}E & F\end{vmatrix}=ACF + \frac{1}{4}BDE - \frac{1}{4}AE^2 - \frac{1}{4}CD^2 - \frac{1}{4}FB^2\)
   
   \(K= \begin{vmatrix} 2A & D \\D & 2F \end{vmatrix} + \begin{vmatrix} 2C & E \\E & 2F \end{vmatrix}= (4AF - D^2) + (4CF - E^2)\)    (Applicable only for Co-incident or Parallel Lines)
   
   OR
   
   \(K= \begin{vmatrix} A & \frac{1}{2}D \\\frac{1}{2}D & F \end{vmatrix} + \begin{vmatrix} C & \frac{1}{2}E \\\frac{1}{2}E & F \end{vmatrix}= (AF - \frac{1}{4}D^2) + (CF - \frac{1}{4}E^2) \)    (Applicable only for Co-incident or Parallel Lines)
   
   Following are the Matrices, Determinants and Related Algebraic Entities used to analyse the General Quadratic Equations in 2 Variables when they are Analysed in form of equation (2)
   
   \(e=\begin{bmatrix} A & B \\B & C \end{bmatrix} ,\hspace{.8cm} J=|e|=\begin{vmatrix} A & B \\B & C \end{vmatrix}= AC - B^2\)
   
   \(E=\begin{bmatrix} A & B & D \\B & C & E \\ D & E & F\end{bmatrix} ,\hspace{.8cm} \Delta=|E|=\begin{vmatrix} A & B & D \\B & C & E \\ D & E & F\end{vmatrix} = ACF + 2BDE - AE^2 - CD^2 - FB^2\)
   
   \(K= \begin{vmatrix} A & D \\D & F \end{vmatrix} + \begin{vmatrix} C & E \\E & F \end{vmatrix}= (AF - D^2) + (CF - E^2)\)    (Applicable only for Co-incident or Parallel Lines)
   
   Please note that All the methods to calculate Determinants given above denoted by same Variable are Equivalent and yeild Similar Results as far as the analysis of the these Quadratic Equations are concerned, although they may evaluate to different actual values.
   
   Along with the above entities, the following expressions help in determining the actual Conic Section represented by any given General Quadratic Equations in 2 Variables
   
   \(M=true\) if \(A=C\) and \(B=0\) otherwise \(M=false\)
   
   \(I=true\) if \(A>0\) and \(C>0\) and \(I=false\) if \(A<0\) and \(C<0\)    (Applicable only for Ellipse/Circle After Ellipse/Hyperbola Type Normalization)
   
   
5. Based on the values that get calculated for the above Matrices, Determinants, Algebraic Entities and Expressions, any given General Quadratic Equation in 2 Variables can found to be representing 1 of the following 11 Geometric Entities
   
   

   S.No.	Geometric Entity	\(\Delta\)	\(J\)	\(M\)	\(I\)	\(K\)
   1.	2 Co-Incident Real Lines	\(=0\)	\(=0\)	NA	NA	\(=0\)
   2.	2 Parallel Real Lines	\(=0\)	\(=0\)	NA	NA	\(<0\)
   3.	2 Parallel Imaginary Lines	\(=0\)	\(=0\)	NA	NA	\(>0\)
   4.	2 Intersecting Real Lines	\(=0\)	\(<0\)	NA	NA	NA
   5.	2 Intersecting Imaginary Lines	\(=0\)	\(>0\)	NA	NA	NA
   6.	Parabola	\(\neq0\)	\(=0\)	NA	NA	NA
   7.	Hyperbola	\(\neq0\)	\(<0\)	NA	NA	NA
   8.	Real Ellipse	\(\neq0\)	\(>0\)	\(false\)	\(true\)	NA
   9.	Imaginary Ellipse	\(\neq0\)	\(>0\)	\(false\)	\(false\)	NA
   10.	Real Circle	\(\neq0\)	\(>0\)	\(true\)	\(true\)	NA
   11.	Imaginary Circle	\(\neq0\)	\(>0\)	\(true\)	\(false\)	NA