# Introduction to Pairs of Lines

Consider two lines

\[\begin{align} {L_1} \equiv y - {m_1}x - {c_1} = 0\\ {L_2} \equiv y - {m_2}x - {c_2} = 0 \end{align}\]

What do you think will \({L_1}{L_2} = 0\) represent? It is obvious that any point lying on \({L_1}\) and \({L_2}\) will satisfy \({L_1}{L_2} = 0,\) and thus \({L_1}{L_2} = 0\) represents the set of points constituting both the lines, i.e.,

\[\boxed{{L_1}{L_2} = 0{\text{ represents the pair of straight lines given by }}{L_1} = 0\,\,{\text{and }}{L_2} = 0}\]

For example, consider the equation \({y^2} - {x^2} = 0.\) What does this represent ? We have

\[\begin{align} \qquad & {y^2} - {x^2} = 0 \qquad \qquad \qquad \qquad...(1)\\ \Rightarrow \qquad & (y + x)(y - x) = 0\\ \Rightarrow \qquad & (1)\;{\rm{represents}}\;{\text{the pair of straight lines}}\;x = y\;{\rm{and}}\;x + y = 0. \end{align}\]

There is nothing special about considering a pair. We can similarly define the joint equation of \(n\) straight lines \({L_i} \equiv y - {m_i}x - {c_i} = 0\;(i = 1,\,2...,n)\) as

\[\begin{align} & {L_1}{L_2}...{L_n} = 0\\ \Rightarrow \qquad & (y - {m_1}x - {c_1})(y - {m_2}x - {c_2})...(y - {m_n}x - {c_n}) = 0 \qquad \qquad \qquad ...(2) \end{align}\]

Any point lying on any of these *n* straight lines will satisfy (2), and thus (2) represents the set of all points constituting the *n* lines, i.e. (2) represents the joint equation of the *n* straight lines.

What is relevant to us at this stage is only a pair of straight lines and it is on a pair of lines that we now focus our attention.