Question

Given a vertex of the square circumscribing the circle |z-1|=√2 as 2+√3i, which of the following is not a vertex of this square ?

(a) (1-√3)+i

(b) –i√3

(c) (√3+i)-i

(d) i√3

The question was asked by my school principal while I was bunking the class.

The above asked question is from Regions in the Complex Plane in chapter Complex Numbers of Engineering Mathematics

Answer 1

Correct choice is (d) i√3

Explanation: The given circle has z0=1 as its center and √2 as radius. Let z1=2+i√3. Now, obtain z2 by rotating z1 anticlockwise by 90^0 about z0 ⇒ z2=(1-√3)+i. Now, z0 is midpoint of z1 and z3 and z2 and z4.

؞(z1+z3)/2 ⇒ (2+i√3+z3)/2=1 ⇒ z3=-i√3 and(z2+z4)/2=z0 ⇒ z4=(√3+i)-i.