Question

Let $a^{x-1}=bc$, $b^{y-1}=ac$ and $c^{z-1}=ab$ then the value of $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$

• A. $2$
• B. $0$
• C. $1$
• D. none of these

Answer 1

Answer: (C)

$1$

Given,

$a^{x-1}=bc$

or, $a^x=abc$

Now taking logarithm with respect to $a$ we get,

$x={\log }_{a}abc$

or, $\frac{1}{x}=\frac{\log a}{\log abc}$......(1).

Also given,

$b^{y-1}=ac$ and $c^{z-1}=ab$ using the aforementioned steps from these two we can get,

$y=\frac{\log b}{\log abc}$......(2) and $z=\frac{\log c}{\log abc}$......(3).

Now adding (1), (2) and (3) we get,

$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{\log a+\log b+\log c}{\log abc}=\frac{\log abc}{\log abc}=1$.