Find the angles between the lines 3 x + y = 1 and x + 3 y = 1 - Vidyadip

Question

Find the angles between the lines $\sqrt 3 x + y = 1$ and $x + \sqrt 3 y = 1$

✓

Answer

We have $\sqrt 3 x + y = 1$
$\Rightarrow y = - \sqrt 3 x + 1$
$\therefore {m_1} = - \sqrt 3$
Also $x + \sqrt 3 y = 1$
$\Rightarrow \sqrt 3 y = - x + 1$
$\Rightarrow y = \frac{{ - 1}}{{\sqrt 3 }}x + \frac{1}{{\sqrt 3 }}$
$\therefore {m_2} = \frac{{ - 1}}{{\sqrt 3 }}$
Let $\theta$ be the angle between the lines. Then,
$\tan \theta = \left| {\frac{{ - \sqrt 3 + \frac{1}{{\sqrt 3 }}}}{{1 + ( - \sqrt 3 )\left( {\frac{{ - 1}}{{\sqrt 3 }}} \right)}}} \right| = \left| {\frac{{\frac{{ - 3 + 1}}{{\sqrt 3 }}}}{{1 + 1}}} \right|$$= \left| {\frac{{ - 2}}{{\sqrt 3 }} \times \frac{1}{2}} \right| = \left| {\frac{{ - 1}}{{\sqrt 3 }}} \right| = \frac{1}{{\sqrt 3 }}$
$\tan \theta = \tan 30^\circ$ and $\tan (180^\circ - 30^\circ )$
$\Rightarrow \theta = 30^\circ$ and 150°

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