The equation of a parabola

 A homogeneous linear system with as many equations 
 as unknowns has a nontrivial  solution if and only 
 if the determinant  of the matrix is zero.

Let us calculate the equation of the parabola passing through points P, Q R:

  c1 y  + c2 x^2 + c3 x + c4 = 0


 This same equation with the points P(x1,y1) Q(x2,y2) and R(x3,y3):

  c1 y1 + c2x1^2 + c3x1 + c4 = 0
  c1 y2 + c2x2^2 + c3x2 + c4 = 0
  c1 y3 + c2x3^2 + c3x3 + c4 = 0
  

 The system of four equations:

  c1 y  + c2 x^2 + c3 x + c4 = 0
  c1 y1 + c2x1^2 + c3x1 + c4 = 0
  c1 y2 + c2x2^2 + c3x2 + c4 = 0
  c1 y3 + c2x3^2 + c3x3 + c4 = 0
 

  The determinant of the system:

    |y    x^2  x   1|
    |y1   x1^2 x1  1| = 0
    |y2   x2^2 x2  1| 
    |y3   x3^2 x3  1|

 The determinant in C language:

    |1    1    1   1|
    |y1   x1^2 x1  1| = 0
    |y2   x2^2 x2  1| 
    |y3   x3^2 x3  1|

To calculate the coefficients of the equation of the parabola, 
we use the cofactor expansion along the first row.
  
  cof(R1,C1) y + cof(R1,C2) x^2 + cof(R1,C3) x + cof(R1,C4) = 0

This equation gives us the equation of the parobola
that passes through the three points  P,  Q and R.