question_answer

If \[{{x}^{a}}{{y}^{b}}={{e}^{m}},\] \[{{x}^{c}}{{y}^{d}}={{e}^{n}},\] \[{{\Delta }_{1}}=\left| \begin{matrix}    m & b  \\    n & d  \\ \end{matrix} \right|,\] \[{{\Delta }_{2}}=\left| \begin{matrix}    a & m  \\    c & n  \\ \end{matrix} \right|\] and \[{{\Delta }_{3}}=\left| \begin{matrix}    a & b  \\    c & d  \\ \end{matrix} \right|,\]  then find values of x and y are respectively.

A)  \[{{\Delta }_{1}}\text{/}{{\Delta }_{3}}\,\,and\,\,{{\Delta }_{2}}\text{/}{{\Delta }_{3}}\]

B)  \[{{\Delta }_{2}}\text{/}{{\Delta }_{1}}\,\,and\,\,{{\Delta }_{3}}\text{/}{{\Delta }_{1}}\]  

C)        \[\log \,({{\Delta }_{1}}\text{/}{{\Delta }_{3}})\,\,and\,\,\log \,({{\Delta }_{2}}\text{/}{{\Delta }_{3}})\]

D)  \[{{e}^{{{\Delta }_{1}}\text{/}{{\Delta }_{3}}}}\,and\,{{e}^{{{\Delta }_{2}}\text{/}{{\Delta }_{3}}}}\]

E)  None of these