Question

Let $z=x+iy$ be a complex number. The equation $arg\left(\frac{z+1}{z}\right)=\frac{\pi }{4}$ represents

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

Answer 1

The correct option is A $x^2+x+y+y^2=0$

$arg\left(\frac{z+1}{z}\right)=\frac{\pi }{4}\Rightarrow arg\left(\frac{x+iy+1}{x+iy}\right)=\frac{\pi }{4}\Rightarrow arg\left(\frac{\left(\right(x+1)+iy)(x-iy)}{x^2+y^2}\right)=\frac{\pi }{4}\Rightarrow arg\left(\frac{x(x+1)+y^2-iy}{x^2+y^2}\right)=\frac{\pi }{4}\Rightarrow {\tan }^{-1}\left(\frac{-y}{x^2+y^2+x}\right)=\frac{\pi }{4}\Rightarrow x^2+y^2+x+y=0$