Question

The numbers of complex numbers z such that $\left| z-i \right|=\left| z-1 \right|=\left| z+1 \right|$ is:
1. $0$
2. $1$
3. $2$
4. Infinite

Answer 1

Hint : To solve this question we start by firstly assuming that $z=x+iy$ (where x stands for real part of the complex number and y stands for the imaginary part of the complex number) and then find out the points equidistant and comparing find out how many complex numbers exist in those points by comparing all three to each other. For this you need to understand how complex numbers work so you can use the logic of complex numbers and its graph, which is called an Argand diagram.

Complete step-by-step answer :
As we saw in hint we first start by assuming a value for z which we can say is
$z=x+iy$
Now since in the question it is given that $\left| z-i \right|=\left| z-1 \right|=\left| z+1 \right|$ , we can put in the value of z in the form of x and y as we know to get
$\left| (x+iy)-i \right|=\left| (x+iy)-1 \right|=\left| (x+iy)+1 \right|$
Taking common $i$ terms together
$\left| x+i(y-1) \right|=\left| (x-1)+iy \right|=\left| (x+1)+iy \right|$
Now since we’re taking the modulus of the complex numbers we get
$\left| z-1 \right|=\left| z+1 \right|$
$\left| (x-1)+iy \right|=\left| (x+1)+iy \right|$
Taking modulus
$\sqrt{{{(x-1)}^{2}}+{{y}^{2}}}=\sqrt{{{(x+1)}^{2}}+{{y}^{2}}}$
Opening brackets and squaring on both sides
${{x}^{2}}+1-2x+{{y}^{2}}={{x}^{2}}+1+2x+{{y}^{2}}$
Cancelling the same terms on both sides we get
$4x=0$
Therefore $x=0$
Now taking the second part of equation that
$\left| z-i \right|=\left| z-1 \right|$
$\left| x+i(y-1) \right|=\left| (x-1)+iy \right|$
Taking modulus on both sides
$\sqrt{{{x}^{2}}-2y+{{y}^{2}}+1}=\sqrt{{{x}^{2}}-2x+1+{{y}^{2}}}$
Putting x as zero as calculated and taking square of both sides
$-2y+{{y}^{2}}+1=1+{{y}^{2}}$
Cancelling same terms on both sides we get
$-2y=0$
Therefore
$y=0$
Therefore we can say that the value of z possible will be $z=(0,0)$ therefore only $1$ value of z is possible
Therefore the answer of the question is the 2 options which is $1$ .
So, the correct answer is “Option 2”.

Note : Argand diagram can be explained as it being the geometric plot of complex numbers as points z where x that is the real part in the complex number and y that is the imaginary part are plotted on the two x and y axis x being the real one and y being the imaginary axis in this situation. The plane where it is plotted is called the argand plane.
Alternate way to solve this question is to find the equidistant point from each part of equation and solving it to find the values possible