Question:In the xy-plane, the circle with equation x^2 + y^2 - 2x + 8y - 8 = 0 has center...

1. TRANSLATE the problem information

• Given information:
  • Circle equation: \(\mathrm{x^2 + y^2 - 2x + 8y - 8 = 0}\)
  • Point \(\mathrm{P(4, -8)}\) lies on the circle
  • Point Q is P rotated 180° about center C
• Need to find: Coordinates of Q

2. INFER the solution strategy

• To find Q after rotation, I first need to locate center C
• Circle equation is in general form - need standard form to identify center
• Once I have center C, I can use the 180-degree rotation relationship

3. SIMPLIFY to find the center by completing the square

• Start with: \(\mathrm{x^2 + y^2 - 2x + 8y - 8 = 0}\)
• Rearrange: \(\mathrm{x^2 - 2x + y^2 + 8y = 8}\)
• Complete the square for x terms: \(\mathrm{x^2 - 2x = (x - 1)^2 - 1}\)
• Complete the square for y terms: \(\mathrm{y^2 + 8y = (y + 4)^2 - 16}\)
• Substitute back: \(\mathrm{(x - 1)^2 - 1 + (y + 4)^2 - 16 = 8}\)
• SIMPLIFY: \(\mathrm{(x - 1)^2 + (y + 4)^2 = 25}\)

This gives us center \(\mathrm{C(1, -4)}\) and radius 5.

4. INFER the rotation relationship

• For a 180-degree rotation about point C, the center becomes the midpoint of the original point P and its image Q
• This means: \(\mathrm{C = (P + Q)/2}\)
• Solving for Q: \(\mathrm{Q = 2C - P}\)

5. SIMPLIFY to find Q's coordinates

\(\mathrm{Q = 2C - P = 2(1, -4) - (4, -8)}\)

\(\mathrm{Q = (2, -8) - (4, -8) = (-2, 0)}\)

Answer: C (-2, 0)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make sign errors when completing the square, particularly with the y terms where \(\mathrm{y^2 + 8y}\) becomes \(\mathrm{(y + 4)^2 - 16}\). A common mistake is writing \(\mathrm{(y - 4)^2}\) instead of \(\mathrm{(y + 4)^2}\), or incorrectly calculating the constant term as -8 instead of -16.

This leads to finding the wrong center coordinates, such as \(\mathrm{C(1, 4)}\) instead of \(\mathrm{C(1, -4)}\), which then produces \(\mathrm{Q = (-2, -16)}\) or similar incorrect coordinates. This typically results in confusion and guessing among the given choices.

Second Most Common Error:

Missing conceptual knowledge about 180-degree rotations: Students may not recognize that after a 180-degree rotation about center C, the center becomes the midpoint between P and Q. Instead, they might attempt to use trigonometric approaches or try to directly apply rotation matrices without understanding the simpler geometric relationship.

This may lead them to select Choice A (-3, 4) or get overwhelmed by the complexity they've created and resort to guessing.

The Bottom Line:

This problem tests whether students can work fluently between different forms of circle equations and understand the geometric meaning of rotations. The key insight is recognizing that completing the square reveals the center, and 180-degree rotations have a simple midpoint relationship.