Question

State True and False

${|z-1|}^2+{|z+1|}^2=4$ represents all points $z$ on the circle with radius $1$ and center at origin.

Answer 1

Let ${|z-1|}^2+{|z+1|}^2=4$ ....(1)
$\therefore {(x-1)}^2+y^2+{(x+1)}^2+y^2=4$ or $2(x^2+y^2+1)=4$
$\therefore x^2+y^2=1$ or ${\left|z\right|}^2=1$
$\therefore \left|z\right|=1$
Thus the equation (1) represents all points $z$ on the circle with centre origin and radius unity.