Question

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

Answer 1

Step1: Calculation the slope of line joining the point $A\left(-2,3\right)$ and $B\left(4,b\right)$

The slope of line passing through point $A\left(-2,3\right)$ and $B\left(4,b\right)$ given by the formula $m_1=\frac{y_2-y_1}{x_2-x_1}$

The given points $A\left(-2,3\right)$ and $B\left(4,b\right)$ is $x_1=-2,x_2=4,y_1=3,y_2=b$.

$$\begin{gathered} \Rightarrow m_1=\frac{b-3}{4-\left(-2\right)} \\ \Rightarrow m_1=\frac{b-3}{4+2} \\ \therefore m_1=\frac{b-3}{6} \end{gathered}$$

`The slope of $AB$ is $m_1=\frac{b-3}{6}$.

Step2: Calculation of slope of line $2x-4y=5$.

The slope-intercept form of the equation of a line is given by the formula $y=mx+c,$ where $m$ is slope and $c$ is the $Y$-intercept of the line.

Simplify the equation $2x-4y=5$ to convert it into slope-intercept form.

$$\begin{gathered} 2x-4y=5 \\ \Rightarrow 4y=2x-5 \\ \Rightarrow y=\frac{1}{2}x-\frac{5}{4}equation\left(1\right) \end{gathered}$$

Comparing equation (1) with equation $y=mx+c$, we get the slope of the line $2x-4y=5$ 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 $2x-4y=5$ .So, the product of their slopes will be equal to $-1$.

$$\begin{gathered} \Rightarrow m_1·m_2=-1 \\ \Rightarrow \frac{b-3}{6}.\frac{1}{2}=-1 \\ \Rightarrow b-3=-12(multiplyingbothsideby12) \\ \therefore b=-9(adding3tobothside) \end{gathered}$$

Hence, the value is $b=-9$.