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

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Solution

Step1: Calculation the slope of line joining the point $A (- 2, 3)$ and $B (4, b)$
The slope of line passing through point $A (- 2, 3)$ and $B (4, b)$ given by the formula $m_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
The given points $A (- 2, 3)$ and $B (4, b)$ is $x_{1} = - 2, x_{2} = 4, y_{1} = 3, y_{2} = b$ .
$\Rightarrow m_{1} = \frac{b - 3}{4 - (- 2)} \Rightarrow m_{1} = \frac{b - 3}{4 + 2} ∴ m_{1} = \frac{b - 3}{6}$
`The slope of $A B$ is $m_{1} = \frac{b - 3}{6}$ .
Step2: Calculation of slope of line $2 x - 4 y = 5$ .
The slope-intercept form of the equation of a line is given by the formula $y = m x + c,$ where $m$ is slope and $c$ is the $Y$ -intercept of the line.
Simplify the equation $2 x - 4 y = 5$ to convert it into slope-intercept form.
$2 x - 4 y = 5 \Rightarrow 4 y = 2 x - 5 \Rightarrow y = \frac{1}{2} x - \frac{5}{4} e q u a t i o n (1)$
Comparing equation (1) with equation $y = m x + c$ , we get the slope of the line $2 x - 4 y = 5$ as $m_{2} = \frac{1}{2}$ .
Step3: Calculation of the value of $b$ .
The lines through $A (- 2, 3)$ and $B (4, b)$ is perpendicular to the line $2 x - 4 y = 5$ .So, the product of their slopes will be equal to $- 1$ .
$\Rightarrow m_{1} \cdot m_{2} = - 1 \Rightarrow \frac{b - 3}{6} . \frac{1}{2} = - 1 \Rightarrow b - 3 = - 12 (m u l t i p l y i n g b o t h s i d e b y 12) ∴ b = - 9 (a d d i n g 3 t o b o t h s i d e)$
Hence, the value is $b = - 9$ .

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