Question

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

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A $0$

${\log }_{\frac{1}{3}}(x+y)+{\log }_3(x-y)=2$

$-{\log }_3(x+y{)}^2+{\log }_3(x-y)=2$

${\log }_3\left(\frac{x-y}{x-y^2}\right)=2$

$\frac{x-y}{x-y^2}=9$

$2^y=2^{9(x+1)}$

$y^2=9x+1$

$x=\frac{y^2-1}{9}$

$x-y=9x-9y^2$

$\frac{y^2-1}{9}-y=\frac{9(y^2-1)}{9}-9y^2$

$\frac{y^2-1-9y}{9}=y^2-1=9y^2$

$y^2-1-9y=9(-1-8y^2)$

$y^2-1-9y=-9-72y^2$

$73y^2+9y-8=0$

Imaginary roots