Question

The locus of the complex number arg(z^* + i√3) - π/4 is equal to?

Answer 1

Let $z=x+iy.$ Hence $x+i(y+\sqrt{3})=\lambda.$

If $\arg(z+i\sqrt{3})=\pi/4,$ then

$$x=y+\sqrt{3}=>y=x-\sqrt{3}$$

The locus of the compex number $z$ is equal to

$$y=x-\sqrt{3}$$

If $\arg(z^*+i\sqrt{3})=\pi/4,$ then $x+i(-y+\sqrt{3})=\lambda.$

$$x=-y+\sqrt{3}=>y=-x+\sqrt{3}$$

The locus of the compex number $z$ is equal to

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