Concurrent Lines
Category : JEE Main & Advanced
Three or more lines are said to be concurrent lines if they meet at a point.
First method : Find the point of intersection of any two lines by solving them simultaneously. If the point satisfies the third equation also, then the given lines are concurrent.
Second method : The three lines \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0,\,\,{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\] and \[{{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}=0\] are concurrent if, \[\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix}\, \right|\,=\,0\].
Third method : The condition for the lines \[P=0,\,\,Q=0\] and \[R=0\] to be concurrent is that three constants \[a,\,\,b,\,\,c\] (not all zero at the same time) can be obtained such that \[aP+bQ+cR=0\].
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