JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Condition for Common Roots

(1) Only one root is common : Let \[\alpha \] be the common root of quadratic equations \[{{a}_{1}}{{x}^{2}}+{{b}_{1}}x+{{c}_{1}}=0\] and \[{{a}_{2}}{{x}^{2}}+{{b}_{2}}x+{{c}_{2}}=0\].

\[\therefore \] \[{{a}_{1}}{{\alpha }^{2}}+{{b}_{1}}\alpha +{{c}_{1}}=0\], \[{{a}_{2}}{{\alpha }^{2}}+{{b}_{2}}\alpha +{{c}_{2}}=0\]

By Crammer’s rule : \[\frac{{{\alpha }^{2}}}{\left| \,\begin{matrix} -{{c}_{1}} & {{b}_{1}}  \\ -{{c}_{2}} & {{b}_{2}}  \\ \end{matrix}\, \right|}=\frac{\alpha }{\left| \,\begin{matrix} {{a}_{1}} & -{{c}_{1}}  \\ {{a}_{2}} & -{{c}_{2}}  \\ \end{matrix}\, \right|}=\frac{1}{\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}}  \\ {{a}_{2}} & {{b}_{2}}  \\ \end{matrix}\, \right|}\]

or \[\frac{{{\alpha }^{2}}}{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}=\frac{\alpha }{{{a}_{2}}{{c}_{1}}-{{a}_{1}}{{c}_{2}}}=\frac{1}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}\]

\[\therefore \] \[\alpha =\frac{{{a}_{2}}{{c}_{1}}-{{a}_{1}}{{c}_{2}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}=\frac{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}{{{a}_{2}}{{c}_{1}}-{{a}_{1}}{{c}_{2}}}\], \[\alpha \ne 0\]

\[\therefore \] The condition for only one root common is

\[{{({{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}})}^{2}}=({{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}})({{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}})\]

(2) Both roots are common: Then required condition is

\[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\]