Question

The number of complex numbers $z$ such that $|z+1|=|z-3|$ equals :-

• A. $0$
• B. $1$
• C. $2$
• D. $\infty$

Answer 1

Answer: (D)

$\infty$

For $z=x+iy$ with $x,y$ are real numbers.

Given,

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

or, $\sqrt{(x+1{)}^2+y^2}=\sqrt{(x-3{)}^2+y^2}$

or, $(x+1{)}^2+y^2=(x-3{)}^2+y^2$

or, $(x+1{)}^2=(x-3{)}^2$

or, $x+1=-(x-3)$ [ Taking the negative sign other we will get some contradiction]

or, $2x=2$

or, $x=1$ $\forall y\in R$.

So it is true for infinitely many value of $y$.

So we've infinitely many $z$.