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: Calculating the slope of line $AB$:
We are given that: $A\left(-2,3\right)$ and $B\left(4,b\right)$
Slope of a line is given by:
$m=\frac{y_2-y_1}{x_2-x_1}$
Thus, slope of line $AB$ is given as:
$m_1=\frac{b-3}{4+2}$
$\Rightarrow m_1=\frac{b-3}{6}$

Step 2: Finding the slope of the given line $2x-4y=5$:
Equation of given line is:
$2x-4y=5$
$\Rightarrow y=\frac{1}{2}x-\frac{5}{4}$
Comparing with $y=mx+c$
$m_2=\frac{1}{2}$ (where $m_2$ is the slope of the given line)

Step 3: Finding the value of ‘$b$’:
We are given that: line $AB$is perpendicular to the given line.
$\Rightarrow m_1\times m_2=-1$
$\Rightarrow \frac{b-3}{6}\times \frac{1}{2}=-1$
$\Rightarrow b-3=-12$
$\Rightarrow b=-9$

Hence, value of $b$ is $-9$