Question

The equation of the line, which bisects the line joining two points $(2,-19)$ and $(6,1)$ and perpendicular to the line joining two points $(-1,3)$ and $(5,-1)$ is:

• A. $3x-2y=30$
• B. $2x-y-3=0$
• C. $2x+3y=20$
• D. None of these

Answer 1

The correct option is A

$3x-2y=30$

Explanation for the correct answer:

Finding the equation of line,

Let the required line be $AB$.

Given, $AB$ bisects the line joining $(2,-19)$ and $(6,1)$.

So, $AB$ will pass through the midpoint of those two points.

The midpoint of $(2,-19)$ and $(6,1)$

$\begin{array}{rcl}\left(x_3+y_3\right)& =& \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ & =& \left(\frac{2+6}{2},\frac{-19+1}{2}\right)\\ & =& \left(4,-9\right)\end{array}$

Given that $AB$ is perpendicular to the line joining $(-1,3)$ and $(5,-1)$.

Slope of $\left(x_4,y_4\right)=(-1,3)$ and $\left(x_5,y_5\right)=(5,-1)$

$\begin{array}{rcl}m& =& \frac{y_5-y_4}{x_5-x_4}\\ & =& \frac{3+1}{-1-5}\\ m& =& -\frac{2}{3}\end{array}$

Therefore, the slope of $AB$:

$\begin{array}{rcl}& =& -\frac{1}{{\displaystyle -\frac{2}{3}}}\\ & =& \frac{3}{2}\end{array}$

Equation of $AB$ is $y-y_3=m\left(x-x_3\right)$

$\begin{array}{rcl}y+9& =& \frac{3}{2}(x-4)\\ & \Rightarrow & 2y+18=3x-12\\ & \Rightarrow & 3x-2y=30\end{array}$

Hence, the correct answer is option (A).