Lab for Determinants and Conic Section Curves

Implicit equation for a line.

Implicit equation for a line.  The general equation for a line in the plane is:

(1)    .  

The coefficients in (1) cannot all be zero.  If it were known a priori which coefficient is non zero, then each term can be divided by it to reduce the number of unknown coefficients to two.  

An alternate way to formulate the solution to (1) is to observe that two additional equations must be satisfied:

(2)      for  i = 1,2.

Equations (1) and (2) form a homogeneous system of three equations in three unknowns.  

Since the solution vector    must be non zero, the determinant of the coefficient matrix must be zero, i.e.

(c) John H. Mathews