Question

Find the number of integral pairs for $(x,y)$ from the following equations
${\log }_{100}|x+y|=\frac{1}{2}$ and ${\log }_{10}y-{\log }_{10}|x|={\log }_{100}4$

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

Answer 1

The correct option is C $2$

$|x+y|=(100{)}^{1/2}|x+y|=10x+y=\pm 10$

${\log }_{10}\left(\frac{y}{|x|}\right)={\log }_{10}2y=2|x|y=\pm 2x$

$x+y=10y=2x3x=10x=\frac{10}{3},y=\frac{20}{3}$

$x+y=10y=-2x-x=10x=-10,y=20$

$x+y=-10y=-2x-x=-10x=10,y=-20$

$x+y=-10y=2x3x=-10x=\frac{-10}{3},y=\frac{-20}{3}$

Number of integer solutions: $(-10,20)$ and $(10,-20)=2$