Question

The acute angle between the lines joining the points $(0,0),(3,2)$ and $(2,2),(3,4)$ is

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

Answer 1

The correct option is B ${\tan }^{-1}\left(\frac{4}{7}\right)$

Let $m_1=$ slope of line joining $(0,0)$ and $(3,2)$
$m_1=\frac{2-0}{3-0}=\frac{2}{3}$
And $m_2=$ slope of line joining $(2,2)$ and $(3,4)$
$m_2=\frac{4-2}{3-2}=2$

Let the acute angle between the lines be $\theta$. Then
$\tan \theta =\left|\frac{m_2-m_1}{1+m_2{m}_1}\right|\Rightarrow \tan \theta =\left|\frac{2-\frac{2}{3}}{1+2\times \frac{2}{3}}\right|\therefore \theta ={\tan }^{-1}\left(\frac{4}{7}\right)$