Question

Find the acute angle between the lines $2x-y+3=0$ and $x+y+2=0$.

Answer 1

Since, slope of line $2x-y+3=0$ is $\frac{-\left(\text{coefficient of x}\right)}{\left(\text{coefficient of y}\right)}=\frac{-2}{-1}=2$

$\therefore m_1=2\dots \dots \left(i\right)$

Slope of line $x+y+2=0$ is $\frac{-\left(\text{coefficient of x}\right)}{\left(\text{coefficient of y}\right)}=\frac{-1}{1}$

$\therefore m_2=-1\dots \dots \left(ii\right)$

Acute angle between lines

$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$

$\theta ={\tan }^{-1}\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$

$={\tan }^{-1}\left|\frac{2-(-1)}{1+\left(2\right)(-1)}\right|$

$={\tan }^{-1}\left|\frac{3}{1-2}\right|={\tan }^{-1}\left|\frac{3}{1}\right|$
$\therefore \theta ={\tan }^{-1}|3|$