The line x - 6y + 11 = 0 bisects the join of (8, -1) and (0, k). Find the value of k.

By mid-point formula,

Mid-point of (8, -1) and (0, k) = $\dfrac{8 + 0}{2}, \dfrac{-1 + k}{2}$

= $\dfrac{8}{2}, \dfrac{k - 1}{2}$

= $\Big(4, \dfrac{k - 1}{2}\Big)$.

Given, line x - 6y + 11 bisects the join of (8, -1) and (0, k).

∴ $\Big(4, \dfrac{k - 1}{2}\Big)$ satisfies the equation x - 6y + 11 = 0.

$\therefore 4 - 6 \times \dfrac{k - 1}{2} + 11 = 0 \\[1em] \Rightarrow 4 - 3(k - 1) + 11 = 0 \\[1em] \Rightarrow 4 - 3k + 3 + 11 = 0 \\[1em] \Rightarrow 18 - 3k = 0 \\[1em] \Rightarrow 3k = 18 \\[1em] \Rightarrow k = \dfrac{18}{3} \\[1em] \Rightarrow k = 6.$

Hence, k = 6.