Introduction to Pair of Lines

1. Any given General Quadratic Equation in 2 Variables mathematically represents a Pair of Lines if the Value of the Determinant of its E-Matrix (denoted by \(\Delta\) ) is Zero.
   
   Depending on the Values of the Determinant of its e-Matrix (denoted by \(J\) ) and the Value of related Algebraic Entity \(K\), the Pair of Lines can be of following 5 Types
   1. 2 Co-Incident Real Lines when \(J=0\) and \(K=0\)
   2. 2 Parallel Real Lines when \(J=0\) and \(K<0\)
   3. 2 Parallel Imaginary Lines when \(J=0\) and \(K>0\)
   4. 2 Intersecting Real Lines when \(J<0\)
   5. 2 Intersecting Imaginary Lines when \(J>0\)
2. A General Quadratic Equation in 2 Variables representing a Pair of Lines can be Factored into it's 2 Constituent Real or Complex Linear Equations as follows
   
   \(A_1x + B_1y + C_1=0\)
   
   \(A_2x + B_2y + C_2=0\)
   
   The Coefficients \(A_1, A_2, B_1\) and \(B_2\) and the Constants \(C_1\) and \(C_2\) can be Real or Complex based on the Type of Pair of Lines as follows
   1. For 2 Co-Incident Real Lines, 2 Parallel Real Lines and 2 Intersecting Real Lines all the Coefficients and Constants are Real.
   2. For 2 Intersecting Imaginary Lines \(A_1\) and \(A_2\), \(B_1\) and \(B_2\), \(C_1\) and \(C_2\) are Complex Conjugates of Each Other.
   3. For 2 Parallel Imaginary Lines \(A_1, A_2, B_1\) and \(B_2\) are Real but \(C_1\) and \(C_2\) are Complex Conjugates of Each Other.
3. The Point of Intersection between 2 Intersecting Imaginary Lines is Real.