If |z|=3 then the points representing -2+4 z lie on (a) the...

If $|z|=3$ then the points representing $-2+4 z$ lie on
(a) the circle of radius 4 and centre at $(-2,0)$
(b) the circle of radius 5 and centre at origin
(c) the circle of radius 12 and centre at $(-2,0)$
(d) the circle passing through the points $(-2,0),(2,0)$ and $(0,2)$

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Key Concepts

Effects of Scaling and Translation

Scaling a complex number by a constant multiplies the distance from the origin by that constant, thereby changing the radius of the original circle. Subsequent translation by adding a constant shifts the circle’s center in the complex plane. Combined, these operations determine the location and size of the resulting circle after the transformation.

Modulus of a Complex Number

The modulus of a complex number, denoted |z|, represents the distance of that number from the origin in the complex plane. When |z| is set to a constant value, it describes a circle centered at the origin with a radius equal to that constant.

Affine Transformations in the Complex Plane

An affine transformation in the complex plane is a function of the form w = az + b, where a and b are complex constants and a is nonzero. Such transformations involve both scaling (by a factor of |a|) and translation (by b), and they map circles to circles, although the center and radius of the circle may change.

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