Question

If $a^x=b^y=c^z$ and $b^2=ac$, then the value of $y$ is equal to

• A. $\frac{2xz}{x+z}$
• B. $\frac{xz}{2(x-z)}$
• C. $\frac{xz}{2(z-x)}$
• D. $\frac{2xz}{(x-z)}$

Answer 1

Answer: (A)

$\frac{2xz}{x+z}$

Let $a^x=b^y=c^z=k$ .....( Given )

$\because a^x=k$

$\Rightarrow a=k^{\frac{1}{x}}$

$\because b^y=k$

$\Rightarrow b=k^{\frac{1}{y}}$ .....$\because c^z=k$

$\Rightarrow c=k^{\frac{1}{z}}$

Given, $b^2=ac$

Substituting values of $a,b,c$ in above equation

${\left(k^{\frac{1}{y}}\right)}^2=k^{\frac{1}{x}}\times k^{\frac{1}{z}}$

$\Rightarrow k^{\frac{2}{y}}=k^{\frac{1}{x}+\frac{1}{z}}$

$\Rightarrow \frac{2}{y}=\frac{1}{x}+\frac{1}{z}$

$\Rightarrow \frac{2}{y}=\frac{x+z}{xz}$

$\Rightarrow y=\frac{2xz}{x+z}$

Option (A) is correct.