GATE 2022 MA – Question 50

Complex Analysis · Conformal Mappings · 2 marks · Multiple select

A Möbius T satisfies T(0)=α,T(α)=0,T(∞)=−α, where α=(−1+i)/√2. L is the line through zero of slope −1; C is the unit circle. Which hold?

T maps L to a line.
T maps L to a circle.
T⁻¹ maps C to a line.
T⁻¹ maps C to a circle.

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Correct answer: (A) T maps L to a line.; (C) T⁻¹ maps C to a line.

Explanation

The unique map is T(z)=α(α−z)/(α+z), with pole −α. Because −α lies on L, its generalized-circle image contains ∞ and is a line. Also T(∞)=−α lies on the unit circle, so its inverse image contains ∞ and is a line. **A,C**.

GATE 2022 MA – Question 50

Explanation

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