Question

If the coordinates of $\mathrm{A}$ and $\mathrm{B}$ be $(1,1)$ and $(5,7)$, then the equation of the perpendicular bisectors the line segment $\mathrm{AB}$ is

• A. $2x+3y=18$
• B. $2\mathrm{x}-3\mathrm{y}+18=0$
• C. $2x+3y-1=0$
• D. $3\mathrm{x}-2\mathrm{y}+1=0$

Answer 1

The correct option is A

$2x+3y=18$

Explanation for the correct option.

Step 1: Find the slope of the line

Given, that a line passing through $(1,1)$ and $(5,7)$,

The slope of the line:

$\begin{array}{rcl}\mathrm{AB}& =& \frac{7-1}{5-1}\\ & =& \frac{3}{2}\end{array}$
The slope of perpendicular line:

$\begin{array}{rcl}\mathrm{AB}& =& \frac{-1}{\left(\frac{3}{2}\right)}\\ & =& \frac{-2}{3}\end{array}$

Mid-point of:

$\begin{array}{rcl}\mathrm{AB}& =& \left(\frac{5+1}{2},\frac{7+1}{2}\right)\\ & =& (3,4)\end{array}$

Step 2: Equation of perpendicular bisector
Therefore, the equation of the perpendicular bisector of $\mathrm{AB}$
$\begin{array}{rcl}y-4& =& \frac{-2}{3}(x-3)\\ 3(y-4)& =& -2(x-3)\\ 3y-12& =& -2x+6\\ 2x+3y-12-6& =& 0\\ 2x+3y-18& =& 0\\ 2x+3y& =& 18\end{array}$

Hence, option A is correct.