Question:Circle A has a diameter with endpoints \((-1, 3)\) and \((5, 3)\).Circle B is obtained by translating circle A down...

1. TRANSLATE the problem information

• Given information:
  • Circle A has diameter endpoints \((-1, 3)\) and \((5, 3)\)
  • Circle B is Circle A translated down 5 units
  • Need to find the equation of Circle B

2. INFER what we need to find first

• To write a circle equation, we need the center and radius
• The center of Circle A is the midpoint of its diameter
• The radius is half the diameter length

3. SIMPLIFY to find Circle A's center

• Using midpoint formula: \(((-1 + 5)/2, (3 + 3)/2) = (2, 3)\)
• Circle A has center \((2, 3)\)

4. SIMPLIFY to find Circle A's radius

• Since both endpoints have y-coordinate 3, the diameter is horizontal
• Diameter length = \(5 - (-1) = 6\)
• Radius = \(6/2 = 3\)

5. TRANSLATE the translation to get Circle B

• "Down 5 units" means subtract 5 from the y-coordinate
• New center: \((2, 3 - 5) = (2, -2)\)
• Radius stays the same: 3

6. SIMPLIFY to write Circle B's equation

• Standard form: \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• Substituting \(\mathrm{h} = 2\), \(\mathrm{k} = -2\), \(\mathrm{r} = 3\):
• \((\mathrm{x} - 2)^2 + (\mathrm{y} - (-2))^2 = 9\)
• Important: \(\mathrm{y} - (-2) = \mathrm{y} + 2\)

Answer: \((\mathrm{x} - 2)^2 + (\mathrm{y} + 2)^2 = 9\), which is Choice B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill when handling negative coordinates: Students correctly find that Circle B has center \((2, -2)\), but when writing the equation, they write \((\mathrm{x} - 2)^2 + (\mathrm{y} - 2)^2 = 9\) instead of \((\mathrm{x} - 2)^2 + (\mathrm{y} + 2)^2 = 9\). They forget that when \(\mathrm{k} = -2\), the standard form \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\) becomes \((\mathrm{x} - 2)^2 + (\mathrm{y} - (-2))^2 = 9\), which simplifies to \((\mathrm{x} - 2)^2 + (\mathrm{y} + 2)^2 = 9\).

This leads them to select Choice D (\((\mathrm{x} - 2)^2 + (\mathrm{y} - 2)^2 = 9\)).

Second Most Common Error:

Poor TRANSLATE reasoning about translation direction: Students might misinterpret "down 5 units" and add 5 to the y-coordinate instead of subtracting, getting center \((2, 8)\) instead of \((2, -2)\).

This leads them to select Choice A (\((\mathrm{x} - 2)^2 + (\mathrm{y} - 8)^2 = 9\)).

The Bottom Line:

This problem tests whether students can systematically work through multiple coordinate geometry concepts while being careful with signs. The key insight is that translations are straightforward coordinate shifts, but writing the final equation requires attention to how negative coordinates appear in standard form.