Question

The equation of the perpendicular bisector of the line segment joining $A(2,3)$ and $B(6,-5)$ is

• A. $x–y=–1$
• B. $x–2y=3$
• C. $x+y=3$
• D. $x-2y=6$

Answer 1

The correct option is D

$x-2y=6$

Explanation for the correct option:

Step 1: Find the slope of the line $AB$

Let $A=\left(2,3\right)=(x_1,y_1)B=(6,-5)=(x_2,y_2)$

Slope of the line $\mathbf{AB}\left(m_1\right)\mathbf{=}\frac{{\mathbf{y}}_{\mathbf{2}}\mathbf{-}{\mathbf{y}}_{\mathbf{1}}}{{\mathbf{x}}_{\mathbf{2}}\mathbf{-}{\mathbf{x}}_{\mathbf{1}}}$

$$\begin{gathered} =\frac{-5-3}{6-2} \\ =-2 \end{gathered}$$

Step 2: Using midpoint formula, find co-ordinates of midpoint of $AB$

Let $M$ be the midpoint of $AB$

$\mathbf{M}\mathbf{=}\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

$\Rightarrow$$M=\left(\frac{2+6}{2},\frac{3-5}{2}\right)$

$\Rightarrow$$M=(4,-1)$

Step 3: Find slope of perpendicular bisector of $AB$

Let $PM$ be the perpendicular bisector of $AB$ with slope $m_2$

As $AB\perp PM$ , $m_1{m}_2=-1$

$\Rightarrow$ $m_2=\frac{1}{2}$

Step 4: Form equation of perpendicular bisector of $AB$

Using slope point form of equation of line

$PM:(y-y_m)=m_2(x-x_m)$

$\Rightarrow$ $y+1=\frac{1}{2}(x-4)$

$\Rightarrow$ $PM:x-2y=6$

Hence option (D) is the correct answer.