The Use of Determinants for Constructing Curves and Surfaces through Specified Points

We describe a technique that uses determinants to
construct lines, circles, and general conic sections
through specified points in the plane. The
procedure is also used to pass planes and spheres
in 3-space through fixed points.
THEOREM: A homogeneous linear system with
as many equations as unknowns has a nontrivial
solution if and only if the determinant of the
coefficient matrix is zero.
We will now show how this result can be used to
determine equations of various curves and
surfaces through specified points.