Question

The number of complex numbers $z$such that $|z-i|=|z-1|=|z+1|$ is

• A. $0$
• B. $1$
• C. $2$
• D. Infinite

Answer 1

Answer: (B)

$1$

$|z-i|=|z-1|=|z+1|$

$Letz=x+iy$

$|x+i(y-1)|=|(x-1)+iy|=|(x+1)+iy|$

$meanspoint(x,y)isequidistantfrom(1,0),(-1,0)and(0,1)$

$andinthiscaseonlyapoint(0,0)$

$existwhichwillbeequidistantfromthesepoints.$

$henceonly1suchcomplex$ $numberexists$