Question

If $a^x=b^y=c^{z}and{b}^2=ac,thenshowthaty=\frac{2zx}{z+x}.$

Answer 1

$Let{a}^x=b^y=c^z=k,thena=k^{\frac{1}{x}},b=k^{\frac{1}{y}}andc=k^{\frac{1}{z}}Now,b^2=ac\Rightarrow (k^{\frac{1}{y}}{)}^2=k^{\frac{1}{x}}.k^{\frac{1}{z}}\Rightarrow k^{\frac{2}{y}}=k^{\frac{1}{x}+\frac{1}{z}}\Rightarrow \frac{2}{y}=\frac{1}{x}+\frac{1}{z}\Rightarrow \frac{2}{y}=\frac{z+x}{xy}\Rightarrow y=\frac{2xy}{z+x}$