Question

measure angle between line y= 2 and √3x + y + 1 = 0 is​

Answer 1

$\begin{gathered}\begin{gathered}\bf \: Given \: equation \: of \: lines \: -   \begin{cases} &\sf{y = 2} \\ &\sf{ \sqrt{3} x + y + 1 = 0} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf  To \: Find :-  \begin{cases} &\sf{angle \: between \: lines}  \end{cases}\end{gathered}\end{gathered}$

Concept used :-

• Let us consider two lines having slope m and M having angle θ between them, then angle 'θ' between the lines is given by

$\boxed{ \blue{ \: \rm : \implies \:tan \theta \: = \bigg|\dfrac{M \: - \: m}{1 + Mm} \bigg| }}$

$\boxed{ \blue{ \: \rm : \implies \:slope \: of \: line \: = \: - \: \dfrac{coefficient \: of \: x}{coefficient \: of \: y} }}$

$\large\underline\purple{\bold{Solution :- }}$

Given lines are

$\rm : \implies \:y \: = \: 2 \: - - (i)$

$\rm : \implies \: \sqrt{3} x + y + 1 = 0 - - (ii)$

Now,

$\boxed{ \blue{ \: \rm : \implies \:Slope \: of \: line \: (i), \: m \: = 0}}$

$\boxed{ \blue{ \: \rm : \implies \:Slope \: of \: line \: (ii), \:M \: = - \sqrt{3} }}$

Now,

$\rm : \implies \:Let \: \theta \: be \: the \: angle \: between \: the \: lines$

So, angle between lines is evaluated as

$\rm : \implies \:tan\theta \: = \: |\dfrac{ - \sqrt{3} - 0}{1 + ( - \sqrt{3}) \times 0 } |$

$\rm : \implies \:tan\theta \: \: = \sqrt{3}$

$\boxed{ \blue{ \: \rm : \implies \:\theta \: = \: \dfrac{\pi}{3} }}$