Question

The points of $z$ satisfying arg $\left(\frac{z-1}{z+1}\right)=\frac{\pi }{4}$ lies on :

• A. an arc of a circle
• B. a parabola
• C. an ellipse
• D. a straight line.

Answer 1

Answer: (A)

an arc of a circle

$arg\left(\frac{z-1}{z+1}\right)=\frac{\pi }{4}$

Let $z=x+iy$

$\Rightarrow arg\left(\frac{x-1+iy}{x+1+iy}\right)=\frac{\pi }{4}$

$\Rightarrow arg\left(\frac{(x-1+iy)(x+1-iy)}{(x+1+iy)(x+1-iy)}\right)=\frac{\pi }{4}$

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

$\Rightarrow arg\left(\frac{x^2-1+y^2+i2y}{(x+1{)}^2+y^2}\right)=\frac{\pi }{4}$

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

$\Rightarrow 1=\frac{2y}{x^2+y^2-1}$

i.e. $x^2+y^2-2y-1=0$ which is an arc of a circle.