Solution:

Area of a triangle $=\frac{1}{2}\left|\begin{array}{lll}\mathrm{x}_{1} & \mathrm{y}_{1} & 1 \\ \mathrm{x}_{2} & \mathrm{y}_{2} & 1 \\ \mathrm{x}_{3} & \mathrm{y}_{3} & 1\end{array}\right|$
$35=\frac{1}{2}\left|\begin{array}{ccc}2 & -6 & 1 \\ 5 & 4 & 1 \\ k & 4 & 1\end{array}\right|$
Expanding with $C_{3}$
$\begin{array}{l}
\rightarrow 70=(20-4 k)-(8+6 k)+(8+30) \\
\rightarrow 70=-10 k+50 \\
\rightarrow 20=-2 k \\
\rightarrow k=-2
\end{array}$