In the xy-plane, circle C has equation \((x - 2)^2 + (y + 3)^2 = 36\). Circle D is the...

1. TRANSLATE the given circle equation

• Given: Circle C has equation \((x - 2)^2 + (y + 3)^2 = 36\)
• This is in standard form \((x - h)^2 + (y - k)^2 = r^2\), which tells us:
  • Center: \((h, k) = (2, -3)\)
  • Radius: \(r = \sqrt{36} = 6\)

2. INFER the reflection transformation

• Reflection across the x-axis follows the rule: \((x, y) \rightarrow (x, -y)\)
• This means:
  • x-coordinate stays the same
  • y-coordinate changes sign
  • Distance properties (like radius) are preserved

3. APPLY CONSTRAINTS to transform the center

• Original center: \((2, -3)\)
• After reflection across x-axis: \((2, -(-3)) = (2, 3)\)
• Radius remains: 6

4. TRANSLATE back to equation form

• Circle D has center \((2, 3)\) and radius 6
• Using standard form: \((x - 2)^2 + (y - 3)^2 = 36\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misidentify the center coordinates from the equation \((x - 2)^2 + (y + 3)^2 = 36\).

They might think the center is \((-2, 3)\) instead of \((2, -3)\), forgetting that \((x - h)^2\) means h is positive 2, and \((y + 3)^2\) means k is negative 3. Starting with the wrong center leads to the wrong reflected center.

This may lead them to select Choice A (\((x + 2)^2 + (y + 3)^2 = 36\)) or Choice D (\((x + 2)^2 + (y - 3)^2 = 36\))

Second Most Common Error:

Poor INFER reasoning: Students incorrectly apply the reflection transformation by changing both coordinates instead of just the y-coordinate.

They might reflect \((2, -3)\) to \((-2, 3)\), thinking reflection across x-axis means "flip everything." This fundamental misunderstanding of what x-axis reflection means causes them to get both coordinates wrong.

This may lead them to select Choice D (\((x + 2)^2 + (y - 3)^2 = 36\))

The Bottom Line:

This problem tests two fundamental skills that work together: reading circle equations correctly and understanding coordinate transformations. Students who struggle often either misread the center coordinates or misunderstand what "reflection across x-axis" actually does to points.