Question

If ${\left(\frac{p}{q}\right)}^{rx-s}={\left(\frac{q}{p}\right)}^{px-q}$, then the value of $x$ is:

• A. $1$
• B. $\frac{q+s}{p+r}$
• C. $\frac{q+r}{q+s}$
• D. $\frac{q+r}{p+s}$

Answer 1

The correct option is B $\frac{q+s}{p+r}$

${\left(\frac{p}{q}\right)}^{rx-s}={\left(\frac{q}{p}\right)}^{px-q}$
$\Rightarrow {\left(\frac{p}{q}\right)}^{rx-s}={\left(\frac{p}{q}\right)}^{-(px-q)}$
$\Rightarrow rx-s=-px+q$

$\Rightarrow rx+px=q+s$
$\Rightarrow x(r+p)=q+s$

$\Rightarrow x=\frac{q+s}{r+p}$