question_answer

The coordinates of two consecutive vertices A and B of a regular hexagon ABCDEF are (1, 0) and (2, 0), respectively. The equation of the diagonal CE is

A)  \[\sqrt{3x}+y=4\]

B)         \[x+\sqrt{3}y+4=0\]

C)  \[x+\sqrt{3}y=4\]        

D)  None of the above

Correct Answer: C

Solution :

AB = 1 \[\Rightarrow \]\[C\equiv \,\,(2+1\times \cos 60{}^\circ ,1\times \sin 60{}^\circ )=\left( \frac{5}{2},\frac{\sqrt{3}}{2} \right)\] \[E\equiv \,\,(1,1\times \sin 60{}^\circ +1\times \sin 60{}^\circ )=(1,\sqrt{3})\] Therefore, the equation of CE is \[y-\sqrt{3}=\frac{\sqrt{3}-\frac{\sqrt{3}}{2}}{1-\frac{5}{2}}(x-1)\Rightarrow x+\sqrt{3}\,\,y=4\]