Question:\(3(\mathrm{x} - 6)^2 + 3(\mathrm{y} + 3)^2 \leq 147\)The graph of the given inequality represents a circular region in the...

1. TRANSLATE the inequality into standard form

• Given: \(3(\mathrm{x} - 6)^2 + 3(\mathrm{y} + 3)^2 \leq 147\)
• To get standard circle form \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 \leq \mathrm{r}^2\), divide everything by 3:
  \((\mathrm{x} - 6)^2 + (\mathrm{y} + 3)^2 \leq 49\)

2. INFER the circle's properties

• From \((\mathrm{x} - 6)^2 + (\mathrm{y} + 3)^2 \leq 49\):
  • Center: \((6, -3)\)
  • Radius: \(\sqrt{49} = 7\)
  • The \(\leq\) symbol means we include points ON and INSIDE the circle

3. INFER the y-coordinate boundaries

• For any point \((\mathrm{a}, \mathrm{b})\) within this region, the y-coordinate b is constrained by the circle's vertical extent
• Minimum possible b = center_y - radius = -3 - 7 = -10
• Maximum possible b = center_y + radius = -3 + 7 = 4
• Valid range: \(-10 \leq \mathrm{b} \leq 4\)

4. APPLY CONSTRAINTS to eliminate impossible values

• Check each answer choice:
  • A. -10: Falls within \([-10, 4]\) → Possible
  • B. -3: Falls within \([-10, 4]\) → Possible
  • C. 4: Falls within \([-10, 4]\) → Possible
  • D. 5: Falls outside \([-10, 4]\) → NOT possible

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to convert \(3(\mathrm{x} - 6)^2 + 3(\mathrm{y} + 3)^2 \leq 147\) into standard circle form, either forgetting to divide by 3 or making algebraic errors during the conversion.

Without the standard form, they can't identify the center and radius correctly, leading to wrong boundary calculations. This causes confusion about what values of b are actually possible within the region.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate INFER reasoning: Students recognize it's a circle but don't connect that the y-coordinate range is determined by center_y ± radius. They might try to substitute specific points or use unnecessarily complex approaches.

Without this key insight, they can't establish the systematic constraint \(-10 \leq \mathrm{b} \leq 4\), making it difficult to eliminate options methodically.

This may lead them to select Choice A (-10) thinking extreme values are automatically impossible.

The Bottom Line:

This problem tests whether students can efficiently extract geometric constraints from algebraic expressions. Success requires translating to standard form AND recognizing how circle geometry constrains coordinate ranges.