Question

Find the equation of the right bisector of the line segment joining the points (a, b) and (a₁, b₁).

Answer 1

Let A (a, b) and B (a₁, b₁) be the given points. Let C be the midpoint of AB.

$\therefore \mathrm{Coordinates}\mathrm{o}fC=\left(\frac{a+a_1}{2},\frac{b+b_1}{2}\right)$
And, slope of AB = $\frac{b_1-b}{a_1-a}$

So, the slope of the right bisector of AB is $-\frac{a_1-a}{b_1-b}$

Thus, the equation of the right bisector of the line segment joining the points (a, b) and (a₁, b₁) is

$$\begin{gathered} y-\frac{b+b_1}{2}=-\frac{a_1-a}{b_1-b}\left(x-\frac{a+a_1}{2}\right) \\ \Rightarrow 2\left(a_1-a\right)x+2y\left(b_1-b\right)+\left(a^2+b^2\right)-\left({a_1}^2+{b_1}^2\right)=0 \\ \mathrm{This}\mathrm{is}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{required}\mathrm{line}. \end{gathered}$$