Question

If $x,y,z$ are in G.P. and $2^{2x}=3^{4z}.e^y,$ then the sum of all possible values of $\frac{x}{y}$ is

• A. ${\log }_{e}2$
• B. ${\log }_{2}e$
• C. $\frac{{\log }_{2}e}{2}$
• D. $\frac{{\log }_{e}2}{2}$

Answer 1

The correct option is C $\frac{{\log }_{2}e}{2}$

$2^{2x}=3^{4z}.e^y\Rightarrow 2x\ln 2=4z\ln 3+y$
and $xz=y^2\Rightarrow \left(\frac{x}{y}\right).\left(\frac{z}{y}\right)=1$
Let $\frac{x}{y}=t\Rightarrow \frac{z}{y}=\frac{1}{t}$
$2t\ln 2=\frac{4}{t}\ln 3+12t^2\ln 2-t-4\ln 3=0$
sum of roots $=\frac{1}{2\ln 2}=\frac{{\log }_{2}e}{2}$