Question

$If{x}^a{y}^b=e^m,x^c{y}^d=e^n,{\Delta }_1=\left|\begin{array}{cc}m& b\\ n& d\end{array}\right|{\Delta }_2=\left|\begin{array}{cc}a& m\\ c& n\end{array}\right|$ and ${\Delta }_3=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|,$then the values of x and y are respectively

• A. $\frac{{\Delta }_1}{{\Delta }_3}$ and $\frac{{\Delta }_2}{{\Delta }_3}$
• B. $\frac{{\Delta }_2}{{\Delta }_1}$ and $\frac{{\Delta }_3}{{\Delta }_1}$
• C. $log\left(\frac{{\Delta }_1}{{\Delta }_3}\right)$ and $log\left(\frac{{\Delta }_2}{{\Delta }_3}\right)$
• D. $e^{\frac{{\Delta }_1}{{\Delta }_3}}$ and $e^{\frac{{\Delta }_2}{{\Delta }_3}}$

Answer 1

The correct option is D $e^{\frac{{\Delta }_1}{{\Delta }_3}}$ and $e^{\frac{{\Delta }_2}{{\Delta }_3}}$

Given $x^a{y}^b=e^m,x^c{y}^d=e^n$
$\Rightarrow alogx+blogy=m$ and $clogx+dlogy=n$
By Cramer’s rule, $logx=\frac{{\Delta }_1}{{\Delta }_3}$ and $logy=\frac{{\Delta }_2}{{\Delta }_3}\Rightarrow x=e^{\frac{{\Delta }_1}{{\Delta }_3}}andy=e^{\frac{{\Delta }_2}{{\Delta }_3}}$.