Lab for Determinants and Conic Section Curves

Implicit equations for parabolas.

Standard equation of a parabola. The standard equation,  ,  of a parabola involves four coefficients:

(5)    .  

The coefficients in (5) 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 three.  Thus, three points    are sufficient to uniquely determine the standard equation of a parabola.  

An alternate way to formulate the solution to (5) is to observe that the three additional equations must be satisfied:

(6)      for  i = 1,2,3.

Equations (5) and (6) form a homogeneous system of four equations in four unknowns.  

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

Alternate equation of a parabola.  The alternate equation    of a parabola involves four coefficients:

(7)    .  

The coefficients in (7) 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 three.  Thus, three points   are sufficient to uniquely determine the alternate equation of a parabola.  

An alternate way to formulate the solution to (7) is to observe that the three additional equations must be satisfied:
   
(8)      for  i = 1,2,3.

Equations (7) and (8) form a homogeneous system of four equations in four unknowns.  

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

(c) John H. Mathews