Circle A has equation \((\mathrm{x} - 7)^2 + (\mathrm{y} + 3)^2 = 1\). In the xy-plane, circle B is obtained...

1. TRANSLATE the circle equation to identify key features

• Given information:
  • Circle A: \((x - 7)² + (y + 3)² = 1\)
  • Need to translate right 4 units

• TRANSLATE the standard form \((x - h)² + (y - k)² = r²\):
  • Center: \((h, k) = (7, -3)\) [Remember: \(y + 3 = y - (-3)\)]
  • Radius: \(r = 1\)

2. TRANSLATE the translation instruction

• "Translate to the right 4 units" means:
  • Add 4 to the x-coordinate of the center
  • Keep y-coordinate and radius unchanged

3. INFER the new center coordinates

• Original center: \((7, -3)\)
• Moving right 4 units: \((7 + 4, -3) = (11, -3)\)
• Radius stays: \(1\)

4. TRANSLATE back to equation form

• New center \((11, -3)\) and radius \(1\) give us:
• \((x - 11)² + (y - (-3))² = 1²\)
• Which simplifies to: \((x - 11)² + (y + 3)² = 1\)

Answer: C. \((x - 11)² + (y + 3)² = 1\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse translation directions and subtract instead of add, or change the wrong coordinate.

For example, thinking "right 4 units" means subtracting 4 from the x-coordinate: \((7 - 4, -3) = (3, -3)\), leading to equation \((x - 3)² + (y + 3)² = 1\).

This may lead them to select Choice B (\((x - 3)² + (y + 3)² = 1\))

Second Most Common Error:

Weak TRANSLATE skill: Students mix up which coordinate changes for horizontal vs vertical translations.

They might incorrectly think moving "right" affects the y-coordinate, changing \(-3\) to \(-3 + 4 = 1\), giving center \((7, 1)\) and equation \((x - 7)² + (y - 1)² = 1\).

This may lead them to select Choice D (\((x - 7)² + (y - 1)² = 1\))

The Bottom Line:

This problem tests your ability to correctly interpret geometric transformations in algebraic form. The key insight is that "right" always means adding to x-coordinates, and translation never changes the size or shape of the figure.