# If $x^ay^b=e^m,x^cy^d=e^n$ $\Delta_1=\begin{vmatrix}m&b\\n&d\end{vmatrix}$,$\Delta_2=\begin{vmatrix}a&m\\c&n\end{vmatrix}$,$\Delta_3=\begin{vmatrix}a&b\\c&d\end{vmatrix}$ then the values of $x$ and $y$ are respectively

$\begin{array}{1 1}(A)\;\large\frac{\Delta_1}{\Delta_3},\frac{\Delta_2}{\Delta_3}&(B)\;\large\frac{\Delta_2}{\Delta_1},\frac{\Delta_3}{\Delta_1}\\(C)\;\log(\Delta_1/\Delta_3),\log(\Delta_2/\Delta_3)&(D)\;e^{\Delta_1/\Delta_3},e^{\Delta_2/\Delta_3}\end{array}$

## 1 Answer

$x^ay^b=e^m,x^cy^d=e^n$
$\Rightarrow a\log x+b\log y=m$
$\Rightarrow c\log x+d\log y=n$
By Cramer's rule we have
$\log x=\large\frac{\Delta_1}{\Delta_3}$
$\log y=\large\frac{\Delta_2}{\Delta_3}$
$\Rightarrow x=e^{\Delta_1/\Delta_3}$
$\Rightarrow y=e^{\Delta_2/\Delta_3}$
Hence (D) is the correct answer.
answered Apr 23, 2014

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