Question

Let z = a + ib (where a, b ∈ R and i = √−1) such that |2z + 3i| = |z²|. Identify the correct statement(s)?

(A) |z|_maximum is equal to 3.

(B) |z|minimum is equal to 1.

(C) If |z| is maximum, then a³ + b³ is equal to 27.

(D) |z| is minimum, then (a² + 2b²) is 2.

Answer 1

The correct option is (D) |z| is minimum, then (a² + 2b²) is 2.

|2z + 3i| ≤ 2 |z| + 3

⇒ 0 ≤ |z| ≤ 3 .........(i)

|2z + 3i| ≥ |2|z| − 3|

|z2| ≥ |2|z| − 3| 

⇒ |z| ≥ 1 ...........(ii)

So, (i) and (ii) gives

1 ≤ |z| ≤ 3

|z|_maximum ⇒ z = 3i

So, a = 0, b = 3 &

|z|_minimum ⇒ z = −i

So, a = 0, b = −1