Find the equation of the right bisector of the line segment joining the points (1, 2) and (5, -6).

Slope of the line joining the points (1, 2) and (5, -6) is,

$\text{m}_1 = \dfrac{-6 - 2}{5 - 1} \\[1em] = -\dfrac{8}{4} \\[1em] = -2.$

Let m₂ be the slope of the right bisector of the above line. Then,

$\Rightarrow m$1 \times m2 = -1 \\[1em] \Rightarrow -2 \times m2 = -1 \\[1em] \Rightarrow m2 = \dfrac{1}{2}.

The mid-point of the line segment joining (1, 2) and (5, -6) will be

$=\Big(\dfrac{1 + 5}{2}, \dfrac{2 + (-6)}{2}\Big) \\[1em] = (3, -2).$

Equation of the line having the slope = $\dfrac{1}{2}$ and passing through (3, -2) can be given by point-slope formula i.e.,

$\Rightarrow y - y$1 = m(x - x1) \\[1em] \Rightarrow y - (-2) = \dfrac{1}{2}(x - 3) \\[1em] \Rightarrow 2(y + 2) = x - 3 \\[1em] \Rightarrow 2y + 4 = x - 3 \\[1em] \Rightarrow x - 2y - 3 - 4 = 0 \\[1em] \Rightarrow x - 2y - 7 = 0.

Hence, the equation of the required right bisector is x - 2y - 7 = 0.