pencil of lines

Let

\displaystyle A_{i}x+B_{i}y+C_{i}=0

(1)

be equations of some lines. Use the short notations $A_{i}x+B_{i}y+C_{i}\,:=\,L_{i}$ .

If the lines $L_{1}=0$ and $L_{2}=0$ have an intersection point $P$ , then, by the parent entry (http://planetmath.org/LineThroughAnIntersectionPoint), the equation

\displaystyle k_{1}L_{1}+k_{2}L_{2}=0

(2)

with various real values of $k_{1}$ and $k_{2}$ can any line passing through the point $P$ ; this set of lines is called a pencil of lines.

Theorem. A necessary and sufficient condition in to three lines

L_{1}=0,\quad L_{2}=0,\quad L_{3}=0

pass through a same point, is that the determinant formed by the coefficients of their equations (1) vanishes:

\left|\begin{matrix}A_{1}&B_{1}&C_{1}\\ A_{2}&B_{2}&C_{2}\\ A_{3}&B_{3}&C_{3}\end{matrix}\right|=\left|\begin{matrix}A_{1}&A_{2}&A_{3}\\ B_{1}&B_{2}&B_{3}\\ C_{1}&C_{2}&C_{3}\end{matrix}\right|=0.

Proof. If the line $L_{3}=0$ belongs to the fan of lines determined by the lines $L_{1}=0$ and $L_{2}=0$ , i.e. all the three lines have a common point, there must be the identity

L_{3}\equiv L_{1}+L_{2},

i.e. there exist three real numbers $k_{1}$ , $k_{2}$ , $k_{3}$ , which are not all zeroes, such that the equation

\displaystyle k_{1}L_{1}+k_{2}L_{2}+k_{3}L_{3}\equiv 0

(3)

is satisfied identically by all real values of $x$ and $y$ . This means that the group of homogeneous linear equations

\displaystyle\begin{cases}k_{1}A_{1}+k_{2}A_{2}+k_{3}A_{3}=0,\\ k_{1}B_{1}+k_{2}B_{2}+k_{3}B_{3}=0,\\ k_{1}C_{1}+k_{2}C_{2}+k_{3}C_{3}=0\end{cases}

has nontrivial solutions $k_{1},\,k_{2},\,k_{3}$ . By linear algebra, it follows that the determinant of this group of equations has to vanish.

Suppose conversely that the determinant vanishes. This implies that the above group of equations has a nontrivial solution $k_{1},\,k_{2},\,k_{3}$ . Thus we can write the identic equation (3). Let e.g. $k_{1}\neq 0$ . Solving (3) for $L_{1}$ yields

L_{1}\equiv-\frac{k_{2}L_{2}+k_{3}L_{3}}{k_{1}},

which shows that the line $L_{1}=0$ belongs to the fan determined by the lines $L_{2}=0$ and $L_{3}=0$ ; so the lines pass through a common point.

References

1 Lauri Pimiä: Analyyttinen geometria. Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).

Title	pencil of lines
Canonical name	PencilOfLines
Date of creation	2013-03-22 18:09:03
Last modified on	2013-03-22 18:09:03
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	8
Author	pahio (2872)
Entry type	Theorem
Classification	msc 51N20
Related topic	LineInThePlane
Related topic	Determinant2
Related topic	HomogeneousLinearProblem
Related topic	Pencil2