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 A1 = (x1, y1) and A2 =(x2, y2) be two fixed points in the plane. Find the equation of the straight line L going through A1 and A2.
Solution Let M= (x, y) be an arbitrary point on L, then one can find three constants a, b and c satisfying
Since A1 and A2 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 A1 and A2 ; 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 A1† =(-1, 2) and A2= =(0,1), then the equation of the line L is in this case:
Application 2 Given three points A1† =(x1, y1), A2† =(x2, y2) and A3† =(x3, y3) in the plan (and not on the same line), find the equation of the circle going through these points.
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 A1 (1, 0), A2 (-1, 2) and A3 (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 ÷37/18 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 Ax2+Bxy+Cy2+Dy+E=0 where A, B, C, D, and E are constants.
2. Keplerís 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 A1 = (x1, y1, z1), A2† =(x2, y2, z2) and A3† =(x3, y3, z3) in the space (and not on the same line). Find the equation of the plane going through these points.
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 A1, A2, A3 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: