Question

The line passing through $A\left(-2,3\right)$and $B\left(4,b\right)$is perpendicular to the line $2x-4y=5$. Find the value of $b$.

Answer 1

Step 1: Finding the slope of the line $AB$:
Given points are $A\left(-2,3\right)$ and $B\left(4,b\right)$
Slope of a line when two points are given is represented by

$m=\frac{y_2-y_1}{x_2-x_1}$
Thus, slope of line $AB$= $m_1=\frac{b-3}{4+2}$
$\Rightarrow m_1=\frac{b-3}{6}$

Step 2: Finding the slope of the line $2x-4y=5$:
The given equation can be written as :
$2x-4y=5$
$\Rightarrow 4y=2x-5$
$\Rightarrow y=\frac{1}{2}x-\frac{5}{4}$
On comparing with $y=mx+c$
Slope , $m_2=\frac{1}{2}$(where $m_2$ is the slope of the line $2x-4y=5$)

Step 3: Finding the value of ‘$b$’ using the condition of perpendicular lines:
When any two lines are given as perpendicular , then
$\Rightarrow m_1.m_2=-1$

$\Rightarrow \frac{b-3}{6}\times \frac{1}{2}=-1$

$\Rightarrow b-3=-12$

$\Rightarrow b=-9$

Hence, value of $b$ is $9$