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

$$\begin{gathered} \mathrm{Here},m=\tan {75}^{\circ } \\ \Rightarrow m=\tan \left({45}^{\circ }+{30}^{\circ }\right) \\ \Rightarrow m=\frac{\tan {45}^{\circ }+\tan {30}^{\circ }}{1-\tan {45}^{\circ }\tan {30}^{\circ }} \\ \Rightarrow m=\frac{1+{\displaystyle \frac{1}{\sqrt{3}}}}{1-{\displaystyle \frac{1}{\sqrt{3}}}}=\frac{\sqrt{3}+1}{\sqrt{3}-1} \\ \Rightarrow m=\frac{\sqrt{3}+1}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}=2+\sqrt{3} \end{gathered}$$

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

$$\begin{gathered} y-2\sqrt{3}=\left(2+\sqrt{3}\right)\left(x-2\right) \\ \Rightarrow y-2\sqrt{3}=\left(2+\sqrt{3}\right)x-4-2\sqrt{3} \\ \Rightarrow \left(2+\sqrt{3}\right)x-y-4=0 \end{gathered}$$