Question

Given the amplitude of the complex number $z-2-3i$ is $\frac{\pi }{4}$, then find the locus of the complex number.

Answer 1

Let $z=x+iy$.

Given the complex number $z-2-3i=(x-2)+i(y-3)$.

Now the amplitude of the above complex number is ${\tan }^{-1}\frac{y-3}{x-2}$.

According to the problem,

${\tan }^{-1}\frac{y-3}{x-2}=\frac{\pi }{4}$

or, $\frac{y-2}{x-2}=1$

or, $y-3=x-2$

or, $x-y+1=0$