Question

The equation $|z-1{|}^2+|z+1{|}^2=4$ represents on the Argand plane

• A. a straight line
• B. an ellipse
• C. a circle with centre origin and radius 2
• D. a circle with centre origin and radius unity

Answer 1

Answer: (D)

We have, $|z-1{|}^2+|z+1{|}^2=4$
$\Rightarrow (x-1{)}^2+y^2+(x+1{)}^2+y^2=4$
(Putting $z=x+iy$ )
$\Rightarrow 2\left(x^2+y^2+1\right)=4$
$\therefore x^2+y^2=1$
or $|z{|}^2=1$
$\Rightarrow |z|=1$
(since $|z|$ cannot be $-ve$ )
Thus, the Eq. (1) represents all points $z$ on the circle with centre origin and radius unity.