Let (2-3 z1 z2)/(3 z1-2 z2) be a point lying inside the...

Let $\frac{2-3 z_{1} z_{2}}{3 z_{1}-2 z_{2}}$ be a point lying inside the circle $|z|=1$ for any two complex numbers $z_{1}$ and $z_{z^{\prime}}$. Then (a) $\mathrm{z}_{1}$ lies inside $|\mathrm{z}|=\frac{2}{3}$ and $\mathrm{z}_{2}$ inside $|\mathrm{z}|=1$ (b) $\mathrm{z}_{1}$ lies inside $|\mathrm{z}|=\frac{2}{3}$ and $\mathrm{z}_{2}$ outside $|\mathrm{z}|=1$ (c) $\mathrm{z}_{1}$ lies outside $|\mathrm{z}|=\frac{2}{3}$ and $\mathrm{z}_{2}$ inside $|\mathrm{z}|=1$ (d) $\mathrm{z}_{1}$ lies outside $|\mathrm{z}|=\frac{2}{3}$ and $\mathrm{z}_{2}$ outside $|\mathrm{z}|=1$

Let $\frac{2-3 z_{1} z_{2}}{3 z_{1}-2 z_{2}}$ be a point lying inside the circle $|z|=1$ for any two complex numbers $z_{1}$ and $z_{z^{\prime}}$. Then
(a) $\mathrm{z}_{1}$ lies inside $|\mathrm{z}|=\frac{2}{3}$ and $\mathrm{z}_{2}$ inside $|\mathrm{z}|=1$
(b) $\mathrm{z}_{1}$ lies inside $|\mathrm{z}|=\frac{2}{3}$ and $\mathrm{z}_{2}$ outside $|\mathrm{z}|=1$
(c) $\mathrm{z}_{1}$ lies outside $|\mathrm{z}|=\frac{2}{3}$ and $\mathrm{z}_{2}$ inside $|\mathrm{z}|=1$
(d) $\mathrm{z}_{1}$ lies outside $|\mathrm{z}|=\frac{2}{3}$ and $\mathrm{z}_{2}$ outside $|\mathrm{z}|=1$

Key Concepts

Complex Numbers

Complex numbers consist of a real part and an imaginary part, and are typically written in the form a + bi where i is the imaginary unit. Understanding complex numbers is fundamental in exploring operations and transformations in the complex plane, as they extend the concept of one-dimensional real numbers to two dimensions.

Modulus and the Unit Circle

The modulus of a complex number, represented as |z|, gives the distance of the point z from the origin in the complex plane. The unit circle, defined by |z| = 1, is a central concept in complex analysis, forming the boundary between points inside (|z| < 1) and outside (|z| > 1) the circle. It is crucial when analyzing the given function's placement within the circle.

Rational Transformations in the Complex Plane

Rational transformations, which involve mappings defined by ratios of polynomial expressions in complex variables, can alter the geometric location of points in the complex plane. They are a general category that includes Möbius transformations, and understanding how these functions map regions (for example, mapping into the unit disk) is key to solving problems where the image of a transformation must lie within a specific region.

Region Mapping and Inequalities

Analyzing which regions in the complex plane map into others under a given function often involves setting up and solving inequalities related to the moduli of complex expressions. This involves determining conditions under which the transformed point satisfies a specific property, such as remaining within the unit circle, and is vital for determining the constraints on the original variables.

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