Question

The locus of the complex number $z=x+iy$ where $i=\sqrt{-1}$ satisfying relation $arg(z-a)=\frac{\pi }{4}$ where $a\in R$ is

• A. $x^2-y^2=a^2$
• B. $x^2+y^2=a^2$
• C. $x+y=a,y>0$
• D. $x-y=a,y>0$

Answer 1

The correct option is D $x-y=a,y>0$

Given that, $z=x+iy$

$\therefore z-a=(x-a)-iy$

$arg\left(z-a\right)={\tan }^{-1}\left(\frac{y}{x-a}\right)$$=\frac{\pi }{4}$

$\frac{y}{x-a}=\tan \left(\frac{\pi }{4}\right)$

$y=x-a$

$\Rightarrow x-y=a\left(y>0\right)$