Question

If $A(-2,1),B(2,3)$ and $C(-2,-4)$ are three points, then the angle between $BA$ and $BC$ is:

• A. ${\tan }^{-1}\left(\frac{3}{2}\right)$
• B. ${\tan }^{-1}\left(\frac{2}{3}\right)$
• C. ${\tan }^{-1}\left(\frac{7}{4}\right)$
• D. $noneofthese$

Answer 1

The correct option is B ${\tan }^{-1}\left(\frac{2}{3}\right)$

Slope of line joining $(x_1,y_1),(x_2,y_2)$ is $m=\frac{y_2-y_1}{x_2-x_1}$

First Consider the side $BA$. Let $m_1$ be the slope of $BA$

$\Rightarrow m_1=\frac{3-1}{2+2}$

$\Rightarrow m_1=\frac{2}{4}$

First Consider the side $BC$. Let $m_2$ be the slope of $BC$

$\Rightarrow m_2=\frac{3+4}{2+2}$

$\Rightarrow m_2=\frac{7}{4}$

Angle between $BA,BC$ is given by $\tan \theta =\frac{m_1-m_2}{1+m_1{m}_2}$

Substituting the values of $m_1,m_2$ we get

$\Rightarrow \tan \theta =\frac{\frac{5}{4}}{1+\frac{14}{16}}$

On solving we get

$\Rightarrow \tan \theta =\frac{2}{3}$

$\Rightarrow \theta ={\tan }^{-1}\left(\frac{2}{3}\right)$