Question

The number of solution pairs $(x,y)$ of the simultaneous equations
${\log }_{1/3}(x+y)+{\log }_3(x-y)=2$
$2^{y^2}={512}^{x+1}$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is B $1$

${\log }_{3^{-1}}(x+y)+{\log }_3(x-y)=2\Rightarrow -{\log }_3(x+y)+{\log }_3(x-y)=2\Rightarrow {\log }_3\left(\frac{x-y}{x+y}\right)=2\Rightarrow \frac{x-y}{x+y}=9\cdots \left(1\right)$

$2^{y^2}=2^{9(x+1)}\Rightarrow y^2=9x+9\cdots \left(2\right)$

Solving $\left(1\right)$ and $\left(2\right),$ we get
$x=15,y=-12$ and
$x=-\frac{15}{16},y=\frac{3}{4}\text{not possible, as}{\log }_3(x-y)\text{is not defined.}$
Hence, there is only one solution pair.