Q71 / 180 · KCET 2024 · Mathematics
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$\tan ^{-1}\left(-\frac{1}{7}\right)$
- Correct answerC$\tan ^{-1}\left(\frac{1}{7}\right)$
- D$\tan ^{-1}\left(\frac{2}{7}\right)$
Solution
$\because y=-x+3 ; m_1=-1=\tan \theta_1$
and $m_2=\frac{y-1}{x-1}=\frac{4-1}{-3-1}$
$\frac{y-1}{x-1}=\frac{3}{-4} ; m_2=\frac{3}{-4}=\tan \theta_2$
Angle between two lines are
$=\tan \left|\theta_2-\theta_1\right|=\frac{\tan \theta_2-\tan \theta_1}{1+\tan \theta_1 \cdot \tan \theta_2}=\frac{m_2 \div m_1}{1+m_1 m_2}$
$=\frac{\frac{-3}{4}+1}{1+\frac{3}{4}}=\frac{1}{4} \times \frac{4}{7}$
Hence, $\left|\theta_2-\theta_1\right|=\tan ^{-1}\left(\frac{1}{7}\right)$
Hence, the correct answer is option: 'C'.