Answer
The equation of the tangent line is $$y=\frac{3}{4} x-\frac{1}{2}$$
Work Step by Step
$$x^{2}-x y-y^{2}=1, \ \ \ \ \ (2,1)$$
Since $x^{2}-x y-y^{2}=1$
by differentiate both sides by $x$, we get
\begin{align} \Rightarrow &2 x-\left(x y^{\prime}+y \cdot 1\right)-2 y
y^{\prime}=0\\
\Rightarrow &2 x-y=x y^{\prime}+2 y y^{\prime}\\
\Rightarrow&
2 x-y=(x+2 y) y^{\prime} \\
\Rightarrow& y^{\prime}=\frac{2 x-y}{x+2 y}
\end{align}
at $(x,y)=(2,1)$, we get $y^{\prime}=\frac{3}{4}. $
so an equation of the tangent line is
$y-1=\frac{3}{4}(x-2)$
i.e
$y=\frac{3}{4} x-\frac{1}{2}$
