Question

Find the shape of locus of point representing the complex numbers $z$ on the Argand diagram for which
${|z-1|}^2+{|z+1|}^2=4$.

• A. Square
• B. Straight Line
• C. Circle
• D. None of these

Answer 1

The correct option is C Circle

Equation $|z-z_1{|}^2+|z-z_2{|}^2=k$ represent a circle if $k\ge \frac{1}{2}|z_1-z_2{|}^2$

Here given equation is $|z-1{|}^2+|z+1{|}^2=4$ where $z_1=1$ and $z_2=-1$ and $k=4$

Now, $\frac{1}{2}|z_1-z_2{|}^2=\frac{1}{2}|1-(-1){|}^2=2$.

Since $k(=4)\ge 2$.

So, the given equation represents a circle.