Question

If $a^x=b^y=c^z$ and $b^2=ac$ then $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in

• A. A.P
• B. G.P
• C. A.G.P
• D. None of these

Answer 1

The correct option is A A.P

$a^x=b^y\Rightarrow a=b^{\frac{y}{x}}$

$b^y=c^z\Rightarrow c=b^{\frac{y}{z}}$

But $ac=b^2$

$\Rightarrow b^{\frac{y}{x}}{b}^{\frac{y}{z}}=b^2$

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

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

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

$\Rightarrow \frac{1}{x}-\frac{1}{y}=\frac{1}{y}-\frac{1}{z}$

$\therefore \frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in A.P.