Question

Find the area enclosed by the curve formed by iz^3+z^2–z+i=0.

(a) π/2

(b) π

(c) 3π/4

(d) 2π

The question was posed to me in an interview.

Origin of the question is Regions in the Complex Plane in section Complex Numbers of Engineering Mathematics

Answer 1

Right option is (b) π

To explain: Dividing the equation by i on both sides, z^3-iz^2+iz+1=0

⇒ z^2(z-i)+i(z-i)=0 ⇒ (z-i)(z^2+i)=0 ⇒ z=i or z^2=-i ⇒ |z|=|i|=1 or |z^2|=|z|^2=|-i|=1

⇒ |z|=1 ⇒ circle of radius 1 is formed. Hence, area=π(1^2)=π.