Question 1209632: What is the area of the region of the complex plane defined by |z| < 3 + 2|z - 1|?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to approach finding the area of the region:
1. **Rewrite in terms of x and y:**
Let z = x + yi, where x and y are real numbers. Then the inequality becomes:
|x + yi| < 3 + 2|x + yi - 1|
√(x² + y²) < 3 + 2√((x - 1)² + y²)
2. **Consider the cases:**
This inequality isn't easy to work with directly. Squaring both sides will remove the square roots, but it will still be messy. Geometrically, |z| represents the distance from the origin, and |z - 1| represents the distance from the point (1, 0). The inequality says the distance from the origin is less than 3 plus twice the distance from (1, 0).
3. **Try some test points:**
* z = 0: |0| < 3 + 2|0 - 1| => 0 < 3 + 2, which is true.
* z = 1: |1| < 3 + 2|1 - 1| => 1 < 3, which is true.
* z = -1: |-1| < 3 + 2|-1 - 1| => 1 < 3 + 4, which is true.
* z = i: |i| < 3 + 2|i - 1| => 1 < 3 + 2√2, which is true.
4. **Consider the boundary:**
The boundary of the region is given by the equation:
√(x² + y²) = 3 + 2√((x - 1)² + y²)
Squaring both sides:
x² + y² = 9 + 12√((x - 1)² + y²) + 4((x - 1)² + y²)
x² + y² = 9 + 12√((x - 1)² + y²) + 4(x² - 2x + 1 + y²)
x² + y² = 9 + 12√((x - 1)² + y²) + 4x² - 8x + 4 + 4y²
-3x² + 8x - 13 - 3y² = 12√((x - 1)² + y²)
Squaring again would lead to a quartic equation, which is generally quite difficult to solve.
5. **Numerical/Graphical Approach:**
Given the difficulty of the algebra, a numerical or graphical approach is recommended. Plot the boundary equation (or inequality) using software like Wolfram Alpha, Desmos, or similar tools. This will give a visual representation of the region.
6. **Approximation:**
The region appears to be a somewhat distorted circle. We can try to estimate the area. The inequality implies that the region is contained within a circle of radius somewhat larger than 3 centered at the origin.
The area is difficult to calculate analytically. A numerical or graphical approach is the most practical way to determine the area.
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