Question

Consider the following subset of the $XY$-plane:
$S = \{(|z - iz|, |z|^2) : z \text{ is a complex number}\}$.
Which one of the following statements is Correct ?

• A. $S$ is a parabola.
• B. $S$ is a circle.
• C. $S$ is an ellipse but not a circle.
• D. $S$ is a hyperbola.

Hint:

Factorizing the term \(z - iz\) as \(z(1 - i)\) simplifies the modulus immediately.
Whenever the relation between the coordinate variables results in a quadratic relation of the form \(x^2 = ky\) or \(y^2 = kx\)., the shape is always a parabola.

Answer 1

The Correct Option is A

To solve this problem, we need to analyze the given subset of the $XY$-plane: \(S = \{ (|z - iz|, |z|^2) : z \text{ is a complex number} \}.\) Here, \(z\ =\ x\ +\ yi\) is a complex number where \(x\) and \(y\) are real.

Let's determine each part of the set expression:

1. The modulus \(|z - iz|\) is calculated as follows: \(|z - iz|\ =\ |(x\ +\ yi)\ -\ i(x\ +\ yi)|\ =\ |x\ +\ yi\ -\ ix\ -\ yi|\ =\ |x\ -\ yi\ +\ yi\ -\ ix|\ =\ |x\ -\ yi|.\) This simplifies to: \(|(x+yi) - i(x+yi)|\ =\ |\sqrt{x^2\ +\ y^2}|\ =\ |x+y|.\)
2. The square of the modulus \(|z|^2\) is given by: \(|z|^2\ =\ (x^2 + y^2).\)

Now we rewrite the set \(S\) as:

\(S = \{ (|x-y|, x^2 + y^2) \}.\)

Let \(u = |x - y|\) and \(v = x^2 + y^2\), then the problem can be analyzed by expressing \(v\) in terms of \(u\) to simplify it:

From \(|x - y| = u\), it represents two cases:

1. For \(x - y = u\):
   Then: \(x^2 + y^2 = v\ = \frac{u^2}{2} + xy\).
2. For \(x - y = -u\):
   Then: \(x^2 + y^2 = v = \frac{u^2}{2} - xy\).

Each form resembles the form of a parabola in the \((u,v)\) coordinate plane. Thus, \(S\) represents a parabola.

Therefore, the correct answer is: $S$ is a parabola.