Question

Find the equation of the line passing through \[(2, 2\sqrt{3})\] and inclined with x-axis at an angle of 75°.

Answer 1

\[\text { Here, } m = \tan {75}^\circ \]

\[ \Rightarrow m = \tan\left( {45}^\circ + {30}^\circ \right)\]

\[ \Rightarrow m = \frac{\tan {45}^\circ + \tan {30}^\circ}{1 - \tan {45}^\circ \tan {30}^\circ}\]

\[ \Rightarrow m = \frac{1 + \frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\]

\[ \Rightarrow m = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \times \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = 2 + \sqrt{3}\]

So, the equation of the line that passes through \[(2, 2\sqrt{3})\] and has slope \[2 + \sqrt{3}\] is

\[y - 2\sqrt{3} = \left( 2 + \sqrt{3} \right)\left( x - 2 \right)\]

\[ \Rightarrow y - 2\sqrt{3} = \left( 2 + \sqrt{3} \right)x - 4 - 2\sqrt{3}\]

\[ \Rightarrow \left( 2 + \sqrt{3} \right)x - y - 4 = 0\]