If the tangent to the curve xy + ax + by = 0 at (1, 1) is inclined at an angle $\tan ^{-1} 2$ with x-axis, then

$a=1, b=-2$

The point (1, 1) lies on the curve xy + ax + by = 0

∴  $a+b=-1$             ........(i)

Now, $x y+a x+b y=0$

$\Rightarrow x \frac{d y}{d x}+y+a+b \frac{d y}{d x}=0$

$\Rightarrow \left(\frac{d y}{d x}\right)_{(1,1)}=-\frac{a+1}{b+1}$       .....(ii)

Since the tangent makes an angle $\tan ^{-1} 2$ with x-axis.

∴ Slope of the tangent = 2

$\Rightarrow 2=-\frac{a+1}{b+1} \Rightarrow a+2 b=-3$          .......(iii)

Solving (i) and (iii), we get a = 1, b = -2