Application
to Geometry

Given some fixed points in the plane or in 3-D space, many problems require finding some geometric figures passing through these points. The examples we are going to see in this page require knowledge of solving linear systems and computing determinants.

** Application 1** Let A

** Solution **Let M= (x, y) be an arbitrary point
on L, then one can find three constants

_{}

Since A_{1 }and A_{2} are on L, one has

_{}

Together with the above equation, we have a **homogeneous
system **in three equations and three variables a, b and c:

_{}

Since we know that there will be a line through A_{1 }and
A_{2 }; this system will have at least one solution *(a, b, c)*.
However, if *(a, b, c)* is a solution, so is *k(a, b, c)* for any
scalar *k *and so the system has infinitely many solutions. Therefore, the
determinant of the coefficient matrix must be zero:

_{}

For example if A_{1
}=(-1, 2) and A_{2= }=(0,1), then the equation of the line
L is in this case:

_{}

or

_{}

** Application 2** Given three points A

** Solution** If M =(x, y) is an arbitrary point on
the circle, then we can write

_{}

where a, b, c and d are constants. Substituting the three
points in the above equation gives the following homogeneous system in four
equations and four variables *a, b, c* and *d*:

_{}

As in Example 1, the system has infinitely many solutions. So, the determinant:

_{}

For example, to find the equation of the circle going
through the points A_{1 }(1, 0), A_{2 }(-1, 2) and A_{3 }(3,
1), we write

_{}

which gives after simplification

_{}

Of course, this can be written as

_{}

The circle has (7/6, 13/6) as center and Ö37/18 as radius.

** Application 3 The Equation of the orbit of a
planet. ** For this application, you
need to know the following

**1.
**The general equation of a conic section in the plane (a
parabola, hyperbola, or ellipse) is given by Ax^{2}+Bxy+Cy^{2}+Dy+E=0
where A, B, C, D, and E are constants.

2.
**Kepler’s first law: **the orbit of an asteroid about the
sun must be an ellipse.

An astronomer who wants to determine the orbit of an asteroid about the sun sets up a Cartesian coordinate system in the plane of the orbit with the sun at the origin. Then the astronomer makes five observations of the position of the asteroid in this system at five different times. This gives five different points on the orbit.

Suppose these points are:

_{}_{}.

Substituting these points in the general equation of a conic section given above, we get the following:

_{}

After expansion and simplification of the determinant, we get the following equation of the orbit:

_{}

** Application 4
**Suppose we have three points A

** Solution **The general equation of a plane is

_{}

Note that as before, multiplying this equation by a any non-zero constant yields an equation of the same plane.

If M =(x, y, z) is an arbitrary point on the plane, then
substituting the coordinates of A_{1}, A_{2}, A_{3} and
M in the above equation gives the following homogeneous system in the variables
_{}:

_{}

Again, the system cannot have a unique solution, so

_{}

For example, the equation of the plane going through the points (1, -1, 3), (0, 1, 7) and (4,0,-1) is

_{}

which gives after simplification:

_{}