Browse Questions

# Find the angle between the lines $y = (2 – \sqrt3 ) (x + 5)$ and $y = (2 + \sqrt3 ) (x – 7).$

Toolbox:
• Angle between two lines is $\theta = \tan^{-1} \bigg| \large\frac{m_1-m_2}{1+m_1m_2} \bigg|$ where $m_1$ and $m_2$ are the slopes of the two lines.
Step 1
The equation of the two lines are
$y=(2-\sqrt 3)(x+5)$---------(1)
$y = (2+\sqrt 3)(x-7)$-----------(2)
This can be written as
$y = (2-\sqrt 3)x+10-5\sqrt 3$
$y=(2+\sqrt 3)x-14-7\sqrt 3$
Hence $m_1=2-\sqrt 3$ and
$\qquad m_2=2+\sqrt 3$
Step 2
The angle between the two lines is
$\therefore \tan \theta = \bigg| \large\frac{(2-\sqrt 3 )-(2+\sqrt 3)}{1+(2-\sqrt 3 )(2+\sqrt 3)} \bigg|$
$= \bigg| \large\frac{-2}{1+(4-3)} \bigg|$
$= \bigg| \large\frac{-2\sqrt3}{2} \bigg|$
$\tan \theta = \sqrt 3$
$\therefore \theta = 60^{\circ}$ or $120^{\circ}$
Hence the angle between the lines is $\large\frac{\pi}{3}$