Question

The angle between the line $x + y = 3$ and the line joining the points $(1, 1)$ and $(-3, 4)$ is:

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

Answer 1

The Correct Option is C

The slope of the line $x + y = 3$ is $-1$.

The slope of the line joining $(1, 1)$ and $(-3, 4)$ is: $m = \frac{4 - 1}{-3 - 1} = \frac{3}{-4} = -\frac{3}{4}$. 

The angle $\theta$ between the two lines is given by: $\tan \theta = \left|\frac{m_1 - m_2}{1 + m_1m_2}\right| = \left|\frac{-1 - (-\frac{3}{4})}{1 + (-1)(-\frac{3}{4})}\right| = \frac{1}{7}$. 

Hence, $\theta = \tan^{-1}\left(\frac{1}{7}\right)$.