Question

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

Answer 1

$Here,m=tan{75}^{\circ }\Rightarrow m=tan({45}^{\circ }+{30}^{\circ })\Rightarrow m=\frac{tan{45}^{\circ }+tan{30}^{\circ }}{1-tan{45}^{\circ }tan{30}^{\circ }}\Rightarrow m=\frac{1+\frac{1}{\sqrt{3}}}{1-\frac{1}{\sqrt{3}}}=\frac{\sqrt{3}+1}{\sqrt{3}-1}\Rightarrow m=\frac{\sqrt{3}+1}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}=2+\sqrt{3}\text{So, the equation of the line that passes through}(2,2\sqrt{3})\text{and has slope}2+\sqrt{3}isy-2\sqrt{3}=(2+\sqrt{3})(x-2)\Rightarrow y-2\sqrt{3}=(2+\sqrt{3})x-4-2\sqrt{3}\Rightarrow (2+\sqrt{3})x-y-4=0$