In the xy-plane, the graph of the given equation x^2 + y^2 - 32y = 0 is a circle. What...

1. INFER what the problem needs

• Given: \(\mathrm{x^2 + y^2 - 32y = 0}\) is a circle equation
• Find: Coordinates of the center
• Strategy: Convert to standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\) to identify center \(\mathrm{(h, k)}\)

2. SIMPLIFY by completing the square for y-terms

• Focus on the y-terms: \(\mathrm{y^2 - 32y}\)
• Complete the square:
  • Take the coefficient of y: \(\mathrm{-32}\)
  • Divide by 2: \(\mathrm{-32/2 = -16}\)
  • Square this result: \(\mathrm{(-16)^2 = 256}\)
  • So \(\mathrm{y^2 - 32y = (y - 16)^2 - 256}\)

3. SIMPLIFY the full equation

• Substitute the completed square back:
  \(\mathrm{x^2 + (y - 16)^2 - 256 = 0}\)
• Rearrange to standard form:
  \(\mathrm{x^2 + (y - 16)^2 = 256}\)

4. INFER the center coordinates

• Compare with standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\):
  • \(\mathrm{(x - 0)^2 + (y - 16)^2 = 256}\)
  • Therefore \(\mathrm{h = 0, k = 16}\)
  • Center is \(\mathrm{(0, 16)}\)

Answer: (B) (0, 16)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when completing the square, particularly getting confused about whether it should be \(\mathrm{(y - 16)^2}\) or \(\mathrm{(y + 16)^2}\).

They might think: "The coefficient is \(\mathrm{-32}\), so I divide by 2 to get \(\mathrm{-16}\), then the completed square is \(\mathrm{(y + 16)^2}\)" - incorrectly thinking the sign stays negative inside the parentheses.

This leads them to write \(\mathrm{x^2 + (y + 16)^2 = 256}\), giving center \(\mathrm{(0, -16)}\).

This may lead them to select Choice (A) (0, -16).

Second Most Common Error:

Inadequate INFER reasoning: Students don't recognize they need to convert to standard form and instead try to find the center directly from the general form.

They might think: "The y-coefficient is \(\mathrm{-32}\), so maybe the center's y-coordinate is related to \(\mathrm{32}\)" and guess that the center is at \(\mathrm{(0, 32)}\).

This may lead them to select Choice (C) (0, 32).

The Bottom Line:

This problem requires careful algebraic manipulation through completing the square, where sign errors are particularly costly and lead directly to wrong answer choices.