The equation \((\mathrm{x} + 1)^2 + \mathrm{y}^2 = 25\) represents circle C. Circle D is obtained by shifting circle C...

1. TRANSLATE the problem information

• Given information:
  • Original circle C: \((\mathrm{x} + 1)^2 + \mathrm{y}^2 = 25\)
  • Circle D is shifted 3 units to the right
• What this tells us: We need to find how a horizontal shift changes the circle's equation

2. INFER the approach needed

• To transform a circle equation, we need to know its center and radius first
• Horizontal shifts only affect the x-coordinate of the center
• The radius stays the same during translations

3. TRANSLATE the original equation to identify key features

• Standard circle form: \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• Rewrite \((\mathrm{x} + 1)^2 + \mathrm{y}^2 = 25\) as \((\mathrm{x} - (-1))^2 + (\mathrm{y} - 0)^2 = 5^2\)
• Center of circle C: \((-1, 0)\) and radius = 5

4. INFER how the transformation affects the center

• "3 units to the right" means add 3 to the x-coordinate
• New center: \((-1 + 3, 0) = (2, 0)\)
• Radius remains: 5

5. SIMPLIFY to write the new equation

• Circle D equation: \((\mathrm{x} - 2)^2 + (\mathrm{y} - 0)^2 = 25\)
• Final form: \((\mathrm{x} - 2)^2 + \mathrm{y}^2 = 25\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "3 units to the right" and subtract 3 instead of adding 3 to the x-coordinate.

They think: "Right means negative direction" or get confused about the sign change from \((\mathrm{x} + 1)\) to the center coordinate. This leads them to calculate the new center as \((-1 - 3, 0) = (-4, 0)\), giving equation \((\mathrm{x} + 4)^2 + \mathrm{y}^2 = 25\).

This may lead them to select Choice B (\((\mathrm{x} + 4)^2 + \mathrm{y}^2 = 25\)).

Second Most Common Error:

Conceptual confusion about transformations: Students confuse horizontal and vertical shifts, thinking "3 units to the right" affects the y-coordinate instead of the x-coordinate.

They might calculate a new center of \((-1, 3)\) or \((-1, -3)\) and create equations like \((\mathrm{x} + 1)^2 + (\mathrm{y} - 3)^2 = 25\).

This may lead them to select Choice C (\((\mathrm{x} + 1)^2 + (\mathrm{y} - 3)^2 = 25\)).

The Bottom Line:

This problem tests whether students can correctly connect transformation language to algebraic operations. The key insight is that "right" in coordinate geometry means "positive x-direction," and transformations affect the center coordinates in a predictable way.