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 with endpoints \((-1, 3)\) and \((5, 3)\)
  • Circle B is Circle A translated down 5 units
  • Need to find equation of Circle B

• What this tells us: We need to find Circle A's center and radius first, then apply the translation.

2. INFER the solution approach

• To write any circle equation, we need center \((\mathrm{h}, \mathrm{k})\) and radius \(\mathrm{r}\)
• Translation only moves the center - radius stays the same
• "Down 5 units" means subtract 5 from the y-coordinate of the center

3. SIMPLIFY to find Circle A's properties

• Center = midpoint of diameter endpoints:
  • x-coordinate: \((-1 + 5)/2 = 4/2 = 2\)
  • y-coordinate: \((3 + 3)/2 = 6/2 = 3\)
  • Center: \((2, 3)\)

• Radius = half the diameter length:
  • Since both endpoints have \(\mathrm{y} = 3\), diameter = \(|5 - (-1)| = 6\)
  • Radius = \(6/2 = 3\)

4. TRANSLATE the transformation to Circle B

• "Down 5 units" means new center: \((2, 3 - 5) = (2, -2)\)
• Radius unchanged: \(\mathrm{r} = 3\)

5. SIMPLIFY to write the equation

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

Answer: Choice B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students misunderstand what "down 5 units" means in the context of the circle equation. They might think this changes the constant term to -5 or affects the x-coordinate instead of the y-coordinate.

When translating the center from \((2, 3)\) to \((2, -2)\), they incorrectly write the equation as \((\mathrm{x} - 2)^2 + (\mathrm{y} - 2)^2 = 9\), thinking the y-coordinate in the equation should match the y-coordinate of the center directly.

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

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when finding the midpoint, particularly with the x-coordinate calculation \((-1 + 5)/2\), getting 3 instead of 2 for the x-coordinate of the center.

This shifts their entire solution and may lead them to select Choice E \(((\mathrm{x} - 3)^2 + (\mathrm{y} + 2)^2 = 9)\).

The Bottom Line:

This problem tests whether students can systematically work through coordinate geometry transformations. The key insight is remembering that in the standard circle equation \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\), when the center moves to \((\mathrm{h}, \mathrm{k}) = (2, -2)\), the equation becomes \((\mathrm{x} - 2)^2 + (\mathrm{y} + 2)^2 = 9\), not \((\mathrm{y} - 2)^2\).