Question

Line through the points $\left(-2,6\right)$ and $\left(4,8\right)$ is perpendicular to the line through the points $\left(8,12\right)$ and $\left(x,24\right)$. Find the value of $x$.

• A. $1$
• B. $4$
• C. $-4$
• D. $2$

Answer 1

The correct option is B

$4$

Explanation for the correct options:

Step 1: Calculate the slope of the first line

For the first line, it is given that,

$\left(x_1,y_1\right)=\left(-2,6\right)$

$\left(x_2,y_2\right)=\left(4,8\right)$

So, using the formula for slope $\mathbf{m}\mathbf{}\mathbf{=}\mathbf{}\frac{{\mathbf{y}}_{\mathbf{2}}\mathbf{-}{\mathbf{y}}_{\mathbf{1}}}{{\mathbf{x}}_{\mathbf{2}}\mathbf{-}{\mathbf{x}}_{\mathbf{1}}}$, the slope $\left(m_1\right)$ of the first line is,

$m_1=\frac{8-6}{4-\left(-2\right)}$

$\Rightarrow$ $m_1=\frac{2}{6}$

$\Rightarrow$ $m_1=\frac{1}{3}$

Step 2: Calculate the slope of the second line

Now, for the second line,

$\left(x_1,y_1\right)=\left(8,12\right)$

$\left(x_2,y_2\right)=\left(x,24\right)$

So, the slope $\left(m_2\right)$ of the second line is,

$m_2=\frac{24-12}{x-8}$

$\Rightarrow$ $m_1=\frac{12}{x-8}$

Step 3: Calculate the value of $x$

Now, we know that, if two lines with slopes $m_1$ and $m_2$ are perpendicular then,

${\mathit{m}}_{\mathbf{1}}\mathbf{·}{\mathit{m}}_{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{-}\mathbf{1}$

According to the question, the given two lines are perpendicular. So, the will follow the above result.

$\frac{12}{x-8}\times \frac{1}{3}=-1$

$\Rightarrow$$\frac{12}{3\left(x-8\right)}=-1$

$\Rightarrow$ $12=-1\times 3\left(x-8\right)$

$\Rightarrow$ $12=-3x+24$

$\Rightarrow$ $3x=24-12$

$\Rightarrow$ $3x=12$

$\Rightarrow$ $x=\frac{12}{3}$

$\Rightarrow$ $x=4$

Hence,option (B) is the correct option.