Let z=x+i y, w=u+ iv where x, y, u, v are real. If w=(2)/(z) and z moves along the circle |z-2 i

step 1 In this question it is given that if z is equal to x plus iota y, w is equals to u plus iota v a

Let z=x+i y, w=u+ iv where x, y, u, v are real. If w=(2)/(z)...

Let $z=x+i y, w=u+$ iv where $x, y, u, v$ are real. If $w=\frac{2}{z}$ and $z$ moves along the circle $|z-2 i|=2 .$ Then the locus of $w .$ (a) perpendicular bisector of the line joining $(0,-1)$ and $(0,0)$ (b) circle with centre $(0,0)$ radius 1 (c) circle with centre $(0,0)$ radius 2 (d) straight line passing through $(0,2)$

Let $z=x+i y, w=u+$ iv where $x, y, u, v$ are real. If $w=\frac{2}{z}$ and $z$ moves along the circle $|z-2 i|=2 .$ Then the locus
of $w .$
(a) perpendicular bisector of the line joining $(0,-1)$ and $(0,0)$
(b) circle with centre $(0,0)$ radius 1
(c) circle with centre $(0,0)$ radius 2
(d) straight line passing through $(0,2)$

Key Concepts

Transformation of Circles

The transformation properties of circles under complex mappings are notable because they often preserve the general shape. Under inversion or related mappings, circles that do not pass through the origin are generally transformed into other circles, while those that do pass through the origin are transformed into straight lines. This principle allows us to determine the locus of transformed points when the original set is a circle.

Locus in the Complex Plane

A locus in the context of complex numbers is the set of all points that satisfy a certain condition or equation. Understanding the concept of a locus is crucial when analyzing the effect of a complex transformation because it enables one to determine the geometric path (such as a circle or line) described by the transformed points.

Complex Inversion

Complex inversion is a mapping of the form w = k/z, which transforms points in the complex plane by taking the reciprocal of their distance from the origin, scaled by a constant. This transformation reverses the positions of points, mapping those near the origin to points far away and vice versa, and is a specific example of a conformal map that preserves angles.

Mobius Transformations

Mobius transformations are a class of functions defined on the complex plane through fractional linear transformations, typically written as w = (az + b)/(cz + d). Although the mapping w = 2/z is a simpler variant, it falls under this broad category. These transformations are conformal (except at singularities) and have the key property of mapping circles and straight lines to circles or lines.

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