Algebra Of Complex Numbers

Z₁ & Z₂ are complex numbers such that;

Z₁≠Z₂ and |Z1| = |Z2|

Z₁ has positive real part &

Z₂ has negative imaginary part
Given,


Now,



is purely imaginary.

The expression represents the location of vertex of a square when the vertices are labeled in anti-clockwise order.

: This term represents a 90° counterclockwise rotation of about the origin. Since the square is labeled anti-clockwise, moving from to involves such a rotation.

: This term represents a translation and scaling of by the complex number , which moves one unit along the real axis and one unit along the imaginary axis from .

By combining these operations — a rotation of and a translation of — we determine the correct location of in the anti-clockwise configuration of a square.

This ensures that lies opposite to in the square and maintains the geometric relationship with .

Therefore, the correct answer is option (D):

The correct answer is option (A): a + a = 0

Let , and .

Now, consider the given condition:

Substituting the values:

This simplifies to a condition that represents a line.

Slope of the line:

Reflection slope:

The line is: , and the reflection also passes through the origin.

Using complex number representation:

Simplifies to: