The points representing the complex number z in the Argand...

The points representing the complex number $z$ in the Argand plane such that $|z|=2$ and $|z-1-i|-|z+1+i|=0$ are
(a) $\pm 2 \mathrm{i}$
(b) $\pm \sqrt{2}(1+\mathrm{i})$
(c) $\pm \sqrt{2}(-1+i)$
(d) $\pm 2$

Key Concepts

Complex Numbers in the Argand Plane

Complex numbers are often represented as points in the Argand plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This geometric representation allows for a visual interpretation of operations such as addition and multiplication, and also simplifies understanding of concepts such as distance and angle between complex numbers.

Modulus of a Complex Number

The modulus of a complex number, denoted as |z|, represents the distance of z from the origin on the Argand plane. It is calculated using the formula |a + bi| = ?(a² + b²). Equations involving the modulus typically represent circles centered at the origin or at other points in the plane when the complex number is translated.

Locus Defined by a Constant Difference of Distances

An equation of the form |z - A| - |z - B| = 0 implies that the distance from z to point A is equal to the distance from z to point B. The set of all points satisfying this condition forms the perpendicular bisector of the line segment joining A and B. This concept is widely used in coordinate geometry to find loci that are equidistant from two fixed points.

Intersection of Loci

Many complex number problems involve finding the intersections of different loci, such as a circle and a line. This often requires combining the geometric definitions—in this case, a circle defined by a constant modulus and a line defined as the perpendicular bisector—to determine the specific points that satisfy both conditions simultaneously.

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