Question

Find the angle between the pair of straight lines $y = (2 - \sqrt 3 )x$ $+ 5$ and $y = (2 + \sqrt 3 )x - 7$.

Answer 1

Hint: Here we will proceed the solution by using the formula of angle between a pair of straight lines.
Here the given pair of straight lines are $y = (2 - \sqrt 3 )x$ $+ 5$ and $y = (2 + \sqrt 3 )x - 7$
If we observe the given lines they are in the form $y = mx + c$ where $m$ slope of the given line.

Complete Step-by-Step Solution:-
So let us consider
$y = (2 - \sqrt 3 )x \to 1$
$y = (2 + \sqrt 3 )x \to 2$
If we compare the given lines with the standard form $y = mx + c$ then
${m_1} = (2 - \sqrt 3 )$ And ${m_2} = (2 + \sqrt 3 )$
We know that angle between pair of straight lines is $\tan \theta = \left| {\dfrac{{{m_2} - {m_1}}}{{1 + {m_2}{m_1}}}} \right|$
Now we know ${m_1}$ and ${m_2}$ values let us substitute in the above formula

$\tan \theta = \left| {\dfrac{{(2 + \sqrt 3 - 2 + \sqrt 3 )}}{{(1 + (4 - 3))}}} \right|$
$\tan \theta = \left| {\dfrac{{2\sqrt 3 }}{2}} \right|$
$\theta = {\tan ^{ - 1}}\sqrt 3$
Therefore angle between given pair of straight lines $\theta = {60^ \circ }$

NOTE: In such type of problems we have to compare given terms with standard terms where we get values for formula to substitute and here in the above problem we know the formula for angle between straight lines and we have compared the given lines with standard form to get values