Question

Two lines passing through the point (2 , 3) intersect each other at an angle of 60° . If slope of one line is 2, then find the equation of the other line ~ note : there are two possible slopes !

Answer 1

`\begin{gathered} \text{Slope of the first line m1=3} \\ let\text{ slope of the other line=m2} \\ The\text{ angle betw}en\text{ two line is 60} \\ \tan 60=|(m1-m2)/(1+m1m2)| \\ √(3)\text =(2-m2)/(1+3m2)| \\ √(3)\text{ =}\pm(2-m2)/(1+3m2) \\ √(3)=\mleft\lbrace(2-m2)/(1+3m2)\text{ }\mright\rbrace\text{or }\sqrt{\text{ 3}}\text{ =-}\mleft\lbrace(2-m2)/(1+3m2)\mright\rbrace \\ √(3)(1+3m2)=(2-m2)\text{ or}\sqrt{\text{ 3}}(1+3m2)\text{ =-(2-m2)} \\ √(3)+3√(3)m2+m2=2\text{ or }\sqrt[]{3}+3\sqrt[]{3}m2-m2=-2 \\ m2=\frac{(2-\sqrt[]{3})}{(2√(3)+1)}\text{ or m2=-}\frac{(2-\sqrt[]{3})}{(2\sqrt[]{3}+1)} \\ The\text{ equation of line passing through (2,3) and having slope }\frac{(2-\sqrt[]{3})}{(2\sqrt[]{3}+1)} \\ y-3=\frac{(2-\sqrt[]{3})}{(2\sqrt[]{3}+1)}(x-2) \\ y(2√(3)+1)-3(2√(3)+1)=(2-√(3))x-2(2-√(3)) \\ (√(3)-2)x+(2\sqrt[]{3}+1)y=-1+8\sqrt[]{3} \\ \text{THis is the equation of other line.} \end{gathered}`