The curve $\mathrm{y}-{\mathrm{e}}^{\mathrm{xy}}+\mathrm{x}=0$  has a vertical tangent at the point

$\mathrm{y}-{\mathrm{e}}^{\mathrm{xy}}+\mathrm{x}=0-\left(1\right)$
Diff (1) w.r.t =x
$\begin{array}{l}\frac{\mathrm{dy}}{\mathrm{dx}}-{\mathrm{e}}^{\mathrm{xy}}\left(\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}\right)+1=0\\ \frac{\mathrm{dy}}{\mathrm{dx}}-{\mathrm{xe}}^{\mathrm{yy}}\frac{\mathrm{dy}}{\mathrm{dx}}-{\mathrm{ye}}^{\mathrm{xy}}+1=0\\ \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{{\mathrm{ye}}^{\mathrm{xy}}-1}{1-{\mathrm{xe}}^{\mathrm{xy}}}\end{array}$
$\because$ tangent is vertical
Slope $\frac{{\mathrm{ye}}^{\mathrm{xy}}-1}{1-{\mathrm{xe}}^{\mathrm{xy}}}=\frac{1}{0}$
$\begin{array}{l}\Rightarrow 1-{\mathrm{xe}}^{\mathrm{xy}}=0\\ {\mathrm{xe}}^{\mathrm{xy}}=1-\left(2\right)\end{array}$
(1, 0) satisfies (1) & (2)