Question

If arg$⟮\frac{z-i}{z+i}⟯$ =$\frac{\pi }{4}$, then z represents a point on

• A. A straight line
• B. A circle
• C. A pair of straight line
• D. None of these

Answer 1

The correct option is B

A circle

arg $⟮\frac{z-i}{z+i}⟯$ = arg$⟮\frac{x+iy-i}{x+iy+i}⟯$ = arg$⟮\frac{x+i(y-1)}{x+i(y+1)}⟯$

= arg$x+i(y-1)$ - arg$x+i(y+1)$

= $ta{n}^{-1}\frac{y-1}{x}-ta{n}^{-1}\frac{y+1}{x}$ = $ta{n}^{-1}\frac{\frac{y-1}{x}-\frac{y+1}{x}}{1+\frac{y-1}{x}.\frac{y+1}{x}}$ = $ta{n}^{-1}\frac{-2x}{x^2+y^2-1}$

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

$\Rightarrow \frac{-2x}{x^2+y^2-1}$ = 1 $\Rightarrow x^2+y^2+2x-1$ = 0

Which represents equation of a circle