Question

If arg $(z-1)$ =$\frac{\pi }{4}$, then the complex number z lies on a

• A. circle
• B. straight line
• C. ellipse
• D. none of these

Answer 1

The correct option is B

straight line

We can solve this problem in two ways. In first method we will replace Z with x+iy and try to establish a relation between x and y with given condition; arg$(z-1)$ = $\frac{\pi }{4}$.
Let $z$ = $x+iy$, then
$z-1$ = $x+iy-1$ = $(x-1)+iy$
arg$(z-1)$ = $ta{n}^{-1}$$\frac{y}{(x-1)}$
= $\frac{\pi }{4}$
$\Rightarrow$ $tan\left(\frac{\pi }{4}\right)$ = $\frac{y}{x-1}$
1 = $\frac{y}{x-1}$
Or $y$ = $x-1$
$\Rightarrow$ Straight line
Second Method
arg$(z-1)$ = $\frac{\pi }{4}$