Question

Let \( T(z) = \frac{az + b}{cz + d}, ad - bc \neq 0 \), be the Möbius transformation which maps the points \( z_1 = 0, z_2 = -i, z_3 = \infty \) in the z-plane onto the points \( w_1 = 10, w_2 = 5 - 5i, w_3 = 5 + 5i \) in the w-plane, respectively. Then the image of the set \( S = \{ z \in \mathbb{C} : \text{Re}(z) < 0 \ \) under the map \( w = T(z) \) is}

• A. \( \{ w \in \mathbb{C} : |w| < 5 \} \)
• B. \( \{ w \in \mathbb{C} : |w| > 5 \} \)
• C. \( \{ w \in \mathbb{C} : |w - 5| < 5 \} \)
• D. \( \{ w \in \mathbb{C} : |w - 5| > 5 \} \)

Hint:

Möbius transformations map half-planes to circular regions. The geometry of the transformation can be used to determine the image of specific sets.

Answer 1

The Correct Option is C

This question involves a Möbius transformation and asks for the image of the set \( S = \{ z \in \mathbb{C} : \text{Re}(z) < 0 \} \) (the left half of the complex plane) under the transformation \( w = T(z) \). We are given that the transformation maps specific points in the z-plane to points in the w-plane. To solve this, we use the fact that Möbius transformations map half-planes to circular regions. - The transformation \( T(z) = \frac{az + b}{cz + d} \) is known to map vertical lines or half-planes in the complex plane to circles or other half-planes. - Since the left half-plane (Re(z) < 0) is mapped to a circular region in the w-plane, the image of the set \( S \) will be a disk centered at \( w = 5 \) with radius 5. Thus, the image of the set \( S \) under \( T(z) \) is \( \{ w \in \mathbb{C} : |w - 5| < 5 \} \), corresponding to option (C).