Question

Interpret the following locii in $zϵC$.

$Arg(z+i)-Arg(z-i)=\pi /2$

• A. semi circle (in the 1st & 4th quadrant) $x^2+y^2=1$
• B. semi circle (in the 2st & 3th quadrant) $x^2+y^2=2$
• C. semi circle (in the 1st & 3rd quadrant) $x^2+y^2=1$
• D. semi circle (in the 2nd & 4th quadrant) $x^2+y^2=2$

Answer 1

The correct option is A semi circle (in the 1st & 4th quadrant) $x^2+y^2=1$

From the above criterion we get
$\frac{z+i}{z-i}=i$

Let $z=x+iy$

$x+iy+i=ix-y+1$

$x=-(y-1)=1-y$ .......(i) and

$ix=i(y+1)$

$x=(y+1)$ ........(ii)

Multiplying i and ii, we get

$x^2=1-y^2$
Or
$x^2+y^2=1$

Now argument is positive, hence it represents the semicircular arc of the circle (1st and 4th quadrant).