# 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.

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 PencilOfLines 2013-03-22 18:09:03 2013-03-22 18:09:03 pahio (2872) pahio (2872) 8 pahio (2872) Theorem msc 51N20 LineInThePlane Determinant2 HomogeneousLinearProblem Pencil2