Let z=x+iy, w=u+iv where x, y, u, v are real. If w=2 iz and z moves such that z=(π)/(4), the loc

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.

Let z=x+iy, w=u+iv where x, y, u, v are real. If w=2 iz and...

Let $\mathrm{z}=\mathrm{x}+\mathrm{iy}, \mathrm{w}=\mathrm{u}+\mathrm{iv}$ where $\mathrm{x}, \mathrm{y}, \mathrm{u}, \mathrm{v}$ are real. If $\mathrm{w}=2 \mathrm{iz}$ and $\mathrm{z}$ moves such that $\arg \mathrm{z}=\frac{\pi}{4}$, the locus of $\mathrm{w}$ is (a) Straight line (b) circle (c) ellipse (d) parabola

Let $\mathrm{z}=\mathrm{x}+\mathrm{iy}, \mathrm{w}=\mathrm{u}+\mathrm{iv}$ where $\mathrm{x}, \mathrm{y}, \mathrm{u}, \mathrm{v}$ are real. If $\mathrm{w}=2 \mathrm{iz}$ and $\mathrm{z}$ moves such that $\arg \mathrm{z}=\frac{\pi}{4}$, the locus of $\mathrm{w}$ is
(a) Straight line
(b) circle
(c) ellipse
(d) parabola

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Key Concepts

Argument of a Complex Number

The argument of a complex number is the angle it makes with the positive real axis. It is critical in describing the direction of the complex number in polar form and is used to define angular conditions or loci in the complex plane.

Complex Multiplication as Transformation

Multiplying complex numbers results in a transformation that consists of rotating and scaling the original number. This concept allows us to map loci of complex numbers from one configuration to another, preserving the structure based on the modulus and argument.

Mapping of Geometric Loci under Complex Functions

This concept involves studying the image of a set of complex numbers under a given complex function. By understanding how multiplication and other functions transform shapes such as lines or curves, one can determine the resulting locus in the complex plane.

Complex Number Representation

A complex number is expressed in the form x + iy, where x is the real part and y is the imaginary part. This representation lays the foundation for understanding operations and transformations in the complex plane, allowing geometric interpretations of algebraic operations.

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