https://file.aiquickdraw.com/mathbot/file/17270395961269xo35uff.jpg

Let's break down the given set and analyze the points.

The set \( S \) is defined as:
\[
S = \{(x, y) : 1 < x^2 + y^2 \leq 4\}
\]
This represents a region between two circles:
- The outer boundary is a circle of radius 2 centered at the origin \( (0,0) \), given by \( x^2 + y^2 = 4 \).
- The inner boundary is a circle of radius 1 centered at the origin, but points within this circle are **excluded** since \( x^2 + y^2 > 1 \).

Now, let's analyze each point based on its relationship with the set \( S \).

### 1. Point \( \left(\sqrt{2}, -\sqrt{2}\right) \)

To check if this point lies within \( S \), calculate \( x^2 + y^2 \):
\[
x^2 + y^2 = \left(\sqrt{2}\right)^2 + \left(-\sqrt{2}\right)^2 = 2 + 2 = 4
\]
Since \( x^2 + y^2 = 4 \), this point lies on the outer boundary of \( S \). Hence, the correct answer is:
- **a. A boundary point of \( S \)**.

### 2. Point \( (0, -1) \)

For this point, we calculate \( x^2 + y^2 \):
\[
x^2 + y^2 = 0^2 + (-1)^2 = 0 + 1 = 1
\]
Since \( x^2 + y^2 = 1 \), this point lies on the **excluded** inner boundary. Therefore, the correct answer is:
- **c. Neither of these** (since it's excluded from \( S \)).

### 3. Point \( (1,1) \)

For this point, calculate \( x^2 + y^2 \):
\[
x^2 + y^2 = 1^2 + 1^2 = 1 + 1 = 2
\]
Since \( 1 < 2 \leq 4 \), this point lies strictly within the region of \( S \), not on the boundary. Therefore, the correct answer is:
- **b. An interior point of \( S \)**.

### Summary of Answers:
1. **a. A boundary point of \( S \)**
2. **c. Neither of these**
3. **b. An interior point of \( S \)**

Let me know if you'd like further details or have any questions!

Here are five related questions to explore:
1. How would the set \( S \) change if the inequality were \( 0 < x^2 + y^2 \leq 4 \)?
2. What is the area of the region defined by \( S \)?
3. Can a point on the inner boundary of \( S \) ever be considered part of the set?
4. How would the analysis change if the set were defined as \( x^2 + y^2 < 4 \)?
5. How do we classify points on the outer boundary of \( S \) in terms of distance from the origin?

**Tip:** For circles, always remember the equation \( x^2 + y^2 = r^2 \), where \( r \) is the radius!

Consider the set S = {(x, y) : 1 < x^2 + y^2 ≤ 4}. Describe the points (√2, -√2), (0, -1), and (1, 1) in relation to the set.

Analyze points in relation to a set defined by 1 < x^2 + y^2 ≤ 4. Learn to classify points as boundary, interior, or neither using key concepts in analytic geometry and inequalities.

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