JEE Main & Advanced Mathematics Straight Line Point of Intersection of Two Lines

Point of intersection of two lines \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\] and \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\] is given by

\[({x}',\,{y}')=\left( \frac{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}\,,\,\frac{{{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}} \right)=\left( \frac{\left| \,\begin{matrix} {{b}_{1}} & {{b}_{2}}  \\ {{c}_{1}} & {{c}_{2}}  \\ \end{matrix}\, \right|}{\left| \,\begin{matrix} {{a}_{1}} & {{a}_{2}}  \\ {{b}_{1}} & {{b}_{2}}  \\ \end{matrix}\, \right|}\,\,,\,\,\frac{\left| \,\begin{matrix} {{c}_{1}} & {{c}_{2}}  \\ {{a}_{1}} & {{a}_{2}}  \\ \end{matrix}\, \right|}{\left| \,\begin{matrix} {{a}_{1}} & {{a}_{2}}  \\ {{b}_{1}} & {{b}_{2}}  \\ \end{matrix}\, \right|} \right)\]