Question

The angle between the line joining the points $\left(1,-2\right),\left(3,2\right)$ and the line $x+2y-7=0$ is

• A. $\mathrm{\pi }$
• B. $\frac{\mathrm{\pi }}{2}$
• C. $\frac{\mathrm{\pi }}{3}$
• D. $\frac{\mathrm{\pi }}{6}$

Answer 1

The correct option is B

$\frac{\mathrm{\pi }}{2}$

Explanation for the correct option:

Step 1: Finding the equation for two point form of line

The two point form of line is expressed as

$y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

The line joining the points are $\left(1,-2\right),\left(3,2\right)$

$$\begin{gathered} \Rightarrow y+2=\frac{2+2}{3-1}\left(x-1\right) \\ \Rightarrow y+2=\frac{4}{2}\left(x-1\right) \\ \Rightarrow y+2=2\left(x-1\right) \\ \Rightarrow y+2=2x-2 \\ \Rightarrow 2x-y-4=0 \end{gathered}$$

Step 2: Finding the slope

Slope of the Equation is Coefficient of $x$ and coefficient of $y$

$\begin{array}{rcl}{m}_1& =& \frac{-2}{-1}\\ & =& 2\end{array}$

Consider the given Equation as,

$x+2y-7=0$

Slope of the Equation is Coefficient of $x$ and coefficient of $y$

$m_2=-\frac{1}{2}$

Step 3: Finding the angle between the line

Then,

$$\begin{gathered} m_1\times m_2=2\times -\frac{1}{2} \\ =-1 \end{gathered}$$

Thus, $m_1$ perpendicular to $m_2$ and the angle between the two lines equal to $\frac{\mathrm{\pi }}{2}\text{or}90°$

Hence, option (B) is the correct answer.