Answer
$y=\frac{1}{2}x+2$
Work Step by Step
$x^{2}+2xy+y^2=12$
$2x+2xy'+2y+8yy'=0$
$2xy'+8yy'=-2x-2y$
$x+4yy'=-x-y$
$y'=-\frac{x+y}{x+4y}$
When $x=2$ and $y=1$, we have $y'=-\frac{2+1}{2+4}=-\frac{1}{2}$, so the equation of the tagent line is $y-1=-\frac{1}{2}(x-2)$, which is $y=-\frac{1}{2}x+2$
