For this problem, we use the formula for the distance between two points:
[tex]d((x_1,y_1),(x_2,y_2))=\sqrt[]{(x_1-x_2)^2+(y_1-y_2)^2}[/tex]Substituting the given points, setting the distance equal to 2√41, and solving for k we get:
[tex]\begin{gathered} 2\sqrt[]{41}=\sqrt[]{(k-7)^2+(-7-3)^2}=\sqrt[]{k^2-14k+49+100} \\ 2\sqrt[]{41}=\sqrt[]{k^2-14k+149} \\ k^2-14k+149=4\cdot41=164 \\ k^2-14k+49=64 \\ (k-7)^2=8^2 \\ k-7=8\text{ or k-7=-8} \\ k=15\text{ or k=-1} \end{gathered}[/tex]