Question

The points $z_1=3+\sqrt{3}i$ and $z_2=2\sqrt{3}+6i$ are given on a complex plane. The complex number lying on the bisector of the angle formed by the vectors $z_1$ and $z_2$ is

• A. $z=\frac{(3+2\sqrt{3})}{2}+\frac{\sqrt{3}+2}{2}i$
• B. $z=5+5i$
• C. $z=-1-i$
• D. none of these

Answer 1

The correct option is B $z=5+5i$

$z_1=3+\sqrt{3}i=(3,\sqrt{3})$
This lies on
$x=\sqrt{3}y$
$y=tan{30}^{0}y$ ...(i)
Similarly
$z_2=2\sqrt{3}+6i=(2\sqrt{3},6)$
This lies on
$y=\sqrt{3}x$
$y=tan{60}^{0}x$ ...(ii)
Hence
The angle bisector will make an angle of
$\frac{{60}^0-{30}^0}{2}+{30}^0$
$={45}^0$
Hence the line will have an equation
$y=tan{45}^{0}x$
$y=x$
The only point lying on $y=x$ from the above
$(5,5)$
$=5+5i$