Question

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. $\sqrt{3}x+y=4$
• B. $x+\sqrt{3}y+4=0$
• C. $x+\sqrt{3}y=4$
• D. None of these

Answer 1

The correct option is C $x+\sqrt{3}y=4$

Refer to the figure attached. $ABCDEF$ is a regular hexagon with side length $1$ with other sides above $x-$ axis. The coordinates of $A$ and $B$ are given.

Length of $AE$ can be found using sine rule in triangle $FEA$

$\frac{\sin {120}^o}{AE}=\frac{\sin {30}^o}{1}$

$\Rightarrow AE=\sqrt{3}$

Since for the given hexagon $AE\perp AB$, we get the coordinates of $E$ as $(1,\sqrt{3})$....(1)

Now, $FC||AB$ and $\angle ECF={30}^o$

Hence, slope of $CE$ is $\tan {120}^o$....(2)

From (1) and (2) equation of $CE$ is $y-\sqrt{3}=\tan {120}^o(x-1)$

Simplifying we get

$\sqrt{3}y+x=4$

Another hexagon can also be formed with vertices other than $A$ and $B$ below x-axis. From symmetry, the slope of that line will be $\tan {30}^o$

But options A and B have negative slopes. Hence, only option C is correct.