Page 842 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

11.1 In this section we describe a technique that uses determinants to construct
lines, circles, and general conic sections through specified points in the plane.
CONSTRUCTING CURVES

AND SURFACES The procedure is also used to pass planes and spheres in 3-space through

THROUGH SPECIFIED fixed points.

POINTS

Prerequisites: Linear Systems

Determinants

Analytic Geometry

The following theorem follows from Theorem 2.3.6.

THEOREM 11.1.1

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 shall now show how this result can be used to determine equations of various curves and surfaces through specified points.

A Line through Two Points

Suppose that and are two distinct points in the plane. There exists a unique line
(1)

that passes through these two points (Figure 11.1.1). Note that , , and are not all zero and that these coefficients are unique
only up to a multiplicative constant. Because and lie on the line, substituting them in 1 gives the two equations

(2)

(3)

Figure 11.1.1
The three equations, 1, 2, and 3, can be grouped together and rewritten as