Pencil of Straight Lines through Intersection of Two Straight Lines

Theorem
Let $u = l_1 x + m_1 y + n_1$.

Let $v = l_2 x + m_2 y + n_2$.

Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, expressed using the general equations:

The pencil of lines through the point of intersection of $\LL_1$ and $\LL_2$ is given by:
 * $\set {u + k v = 0: k \in \R} \cup \set {\LL_2}$

Proof
Let $\LL$ denote an arbitrary straight line through the point of intersection of $\LL_1$ and $\LL_2$.

From Equation of Straight Line through Intersection of Two Straight Lines, $\LL$ can be given by an equation of the form:


 * $u + k v = 0$

It remains to be seen that the complete pencil of lines through the point of intersection of $\LL_1$ and $\LL_2$ can be obtained by varying $k$ over the complete set of real numbers $\R$.

We have that $\LL$ can also be given by:


 * $\paren {l_1 x + m_1 y + n_1} - k \paren {l_2 x + m_2 y + n_2} = 0$

That is:
 * $\paren {l_1 - k l_2} x + \paren {m_1 - k m_2} y + \paren {n_1 - k n_2} = 0$

Let the slope of $\LL$ be $\tan \psi$ where $\psi$ is the angle $\LL$ makes with the $x$-axis.