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Home  >>  CBSE XI  >>  Math  >>  Straight Lines
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Find the angle between the lines $y = (2 – \sqrt3 ) (x + 5)$ and $y = (2 + \sqrt3 ) (x – 7).$

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  • 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}$
answered Jun 30, 2014 by thanvigandhi_1
 

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