Concurrency of Three Lines

  \(\textbf{Art 8:  } \qquad\boxed{{\text{Concurrency}}}\)

Consider three different straight lines L₁, L₂ and L₃:

\[\begin{align}
  & {{L}_{1}}\equiv {{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\qquad \qquad ...\text{ }\left( 1 \right) \\ 
 & {{L}_{2}}\equiv {{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\qquad \qquad ...\text{ }\left( 2 \right) \\ 
 & {{L}_{3}}\equiv {{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}=0\qquad \qquad ...\text{ }\left( 3 \right) \\ 
\end{align}\]

We need to evaluate the constraint on the coefficients   \(a_{i}^{\mathbf{'}}s,\,\,b_{i}^{\mathbf{'}}s\,\,\text{and}\,\,c_{i}^{\mathbf{'}}s\) such that the three lines are concurrent.

Let us first determine the point P of intersection of L₁ and L₂. By Art - 5, it will be

\[P\equiv \frac{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}},\,\,\frac{{{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}\]

Thus three lines will be concurrent if L₃ passes through P too, that is P satisfies the equation of L₃. Thus,

\[\begin{align}& \qquad\qquad{{a}_{3}}\left( \frac{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}} \right)+\,{{b}_{3}}\left( \frac{{{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}} \right)+{{c}_{3}}=0 \\ 
 & \Rightarrow\qquad {{a}_{3}}\left( {{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}} \right)+{{b}_{3}}\left( {{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}} \right)+{{c}_{3}}\left( {{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}} \right)=0 \\ 
 & \Rightarrow \qquad{{a}_{1}}\left( {{b}_{2}}{{c}_{3}}-{{b}_{3}}{{c}_{2}} \right)+{{b}_{1}}\left( {{c}_{2}}{{a}_{3}}-{{c}_{3}}{{a}_{2}} \right)+{{c}_{1}}\left( {{a}_{2}}{{b}_{3}}-{{a}_{3}}{{b}_{2}} \right)=0 \\ 
\end{align}\]

This relation can be written compactly in determinant form as

\[\left| \begin{matrix}
   {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\
   {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\
   {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\
\end{matrix} \right|=0\]

This is the condition that must be satisfied for the three lines to be concurrent.

For example, consider the three lines \(2x-3y+5=0,3x+4y-7=0\,and\,\,9x-5y+8=0\). These three lines are concurrent because the determinant of the coefficients is 0, i.e,

\[\left| \begin{matrix}
   2 & -3 & 5  \\
   3 & 4 & -7  \\
   9 & -5 & 8  \\
\end{matrix} \right|=0\]