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 ?
Show 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.
• Step 1 : Understanding the Question:
We are given a subset \(S\) of the \(XY\)-plane defined parametrically by a complex number \(z\).
Specifically, the coordinates \((x, y)\) of points in \(S\) are given by \(x = |z - iz|\) and \(y = |z|^2\).
We need to determine the geometric curve represented by this set of points.
• Step 2 : Key Formula or Approach:
We can simplify the expression for \(x\) by factoring out the complex number \(z\).
Recall the property of modulus of complex numbers: \(|w_1 w_2| = |w_1| |w_2|\).
Once both \(x\) and \(y\) are written in terms of \(|z|\)., we can eliminate \(|z|\) to find the direct algebraic relationship between \(x\) and \(y\).
• Step 3 : Detailed Explanation:
Let us write the coordinates of a point \((x, y) \in S\):
\[ x = |z - iz| \]
\[ y = |z|^2 \]
Let us simplify the expression for \(x\):
\[ z - iz = z(1 - i) \]
Using the multiplicative property of the modulus:
\[ x = |z(1 - i)| = |z| \cdot |1 - i| \]
The modulus of the complex number \(1 - i\) is:
\[ |1 - i| = \sqrt{1^2 + (-1)^2} = \sqrt{2} \]
Therefore:
\[ x = \sqrt{2} |z| \]
From this, we can express \(|z|\) as:
\[ |z| = \frac{x}{\sqrt{2}} \]
Now, substitute this expression for \(|z|\) into the equation for \(y\):
\[ y = |z|^2 = \left( \frac{x}{\sqrt{2}} \right)^2 = \frac{x^2}{2} \]
This simplifies to:
\[ x^2 = 2y \]
Since \(x = \sqrt{2}|z| \ge 0\)., this represents the right half of the parabola \(x^2 = 2y\).
The locus of points is a parabola.
• Step 4 : Final Answer:
The set \(S\) represents a parabola.
This corresponds to option (A).
