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.

\[\therefore\text {  Coordinates of C } = \left( \frac{a + a_1}{2}, \frac{b + b_1}{2} \right)\]

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

\[y - \frac{b + b_1}{2} = - \frac{a_1 - a}{b_1 - b}\left( x - \frac{a + a_1}{2} \right)\]

\[ \Rightarrow 2\left( a_1 - a \right)x + 2y\left( b_1 - b \right) + \left( a^2 + b^2 \right) - \left( {a_1}^2 + {b_1}^2 \right) = 0 \]

This is equation of the required line .