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

We have, $z = x + iy $ and $ arg (\frac{z+1}{z} )= \frac{\pi}{4}$
$\because\frac{ z+1}{z} = \frac{x +iy +1}{x+iy} $
$= \frac{\left(\left(x+1\right)+iy\right)\left(x -iy\right)}{x^{2}+y^{2}} $
$ \Rightarrow \frac{z+1}{z} = \frac{x^{2}+x+y^{2}+ \left(xy -xy -y\right)i}{x^{2} +y^{2}} $
$ =\frac{ x^{2}+y^{2} +x -yi}{x^{2} + y^{2}} $
$ \therefore arg \left(\frac{z+1}{z}\right) = tan^{-1} \left(\frac{-y}{x^{2} +y^{2} +x}\right) = \frac{\pi}{4} $
$ -\frac{y}{x^{2}+y^{2} +x} = tan \frac{\pi}{4} = 1 $
$ \Rightarrow -y = x^{2} +y^{2} + x $
$ \Rightarrow x^{2} +y^{2} + x +y = 0$