---
title: "Find a bilinear transformation"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:find-a-bilinear-transformation"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/find-a-bilinear-transformation
site: BCT Engineering
---

# Find a bilinear transformation

Mathematics III · Chapter 2: Line, Surface and Volume Integrals

- Asked in: 2082 Chaitra (new course), 2 marks
- Video: [Bilinear Transformation |  z = 0, -1, i to w = i, 0, ∞ | Image of Unit Circle | Applied Mathematics](https://www.youtube.com/watch?v=DCmB4xWPOQQ) — Engineering Exam Hub
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/find-a-bilinear-transformation
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Find the bilinear transformation which maps the points z = 0, −1, i into w = i, 0, ∞.

## Short answer

The bilinear transformation is w = (z+1)/(z-i). The numerator factor (z+1) comes from z=-1 mapping to 0; the denominator factor (z-i) from z=i mapping to ∞. The constant k=1 is fixed by z=0 → i.

## Answer

Let the bilinear transformation be w = (az + b)/(cz + d).

- z = −1 maps to w = 0, so the numerator has a factor (z + 1).
- z = i maps to w = ∞, so the denominator has a factor (z − i).

Hence w = k (z + 1)/(z − i).

Use z = 0 → w = i:

```text
i = k (1)/(−i) = k · i      (since 1/(−i) = i)
k = 1
```

**Answer: w = (z + 1)/(z − i).**

Check: z = 0 gives 1/(−i) = i; z = −1 gives 0; z = i makes the denominator zero, so w = ∞.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. z = -1 -> w = 0 implies the numerator has factor:
   A. (z - 1)
   B. (z + 1)
   C. (z - i)
   D. z
   Answer: B. w is 0 when numerator is 0. (Key point: Zero)

2. z = i -> w = infinity implies the denominator has factor:
   A. (z + i)
   B. (z + 1)
   C. (z - i)
   D. (z - 1)
   Answer: C. Denominator is 0 at z = i. (Key point: Pole)

3. Using z=0 -> w=i gives k:
   A. -1
   B. i
   C. -i
   D. 1
   Answer: D. i = k/(-i) = k i, so k = 1. (Key point: Constant)

## Related in Mathematics III

- [Find a bilinear transformation](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Afind-a-bilinear-transformation-two) — Mathematics III, chapter 2

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