The equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle? x^2...

1. INFER what approach is needed

• Given: \(\mathrm{x^2 + 20x + y^2 + 16y = -20}\)
• Goal: Find center coordinates
• Strategy: The equation is in general form, but I need standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\) to read the center \(\mathrm{(h, k)}\)
• This means I need to complete the square for both x and y terms

2. SIMPLIFY by completing the square for x terms

• Start with: \(\mathrm{x^2 + 20x}\)
• Take half the coefficient of x: \(\mathrm{20 \div 2 = 10}\)
• Square it: \(\mathrm{10^2 = 100}\)
• So \(\mathrm{x^2 + 20x + 100 = (x + 10)^2}\)

3. SIMPLIFY by completing the square for y terms

• Start with: \(\mathrm{y^2 + 16y}\)
• Take half the coefficient of y: \(\mathrm{16 \div 2 = 8}\)
• Square it: \(\mathrm{8^2 = 64}\)
• So \(\mathrm{y^2 + 16y + 64 = (y + 8)^2}\)

4. SIMPLIFY the equation by adding the completing square terms to both sides

• Need to add \(\mathrm{100 + 64 = 164}\) to both sides:
• \(\mathrm{x^2 + 20x + 100 + y^2 + 16y + 64 = -20 + 164}\)
• \(\mathrm{(x + 10)^2 + (y + 8)^2 = 144}\)

5. TRANSLATE from standard form to center coordinates

• Standard form is \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\) where center is \(\mathrm{(h, k)}\)
• I have \(\mathrm{(x + 10)^2 + (y + 8)^2 = 144}\)
• Rewrite as \(\mathrm{(x - (-10))^2 + (y - (-8))^2 = 144}\)
• Therefore: \(\mathrm{h = -10}\), \(\mathrm{k = -8}\)

Answer: B. (-10, -8)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students correctly complete the square to get \(\mathrm{(x + 10)^2 + (y + 8)^2 = 144}\), but then incorrectly identify the center as \(\mathrm{(10, 8)}\) instead of \(\mathrm{(-10, -8)}\).

They forget that the standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\) means the center is \(\mathrm{(h, k)}\), so when they see \(\mathrm{(x + 10)^2}\), they need to recognize this as \(\mathrm{(x - (-10))^2}\), making \(\mathrm{h = -10}\).

This may lead them to select Choice C (10, 8).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make arithmetic errors when completing the square, such as incorrectly calculating \(\mathrm{(20/2)^2 = 100}\) or \(\mathrm{(16/2)^2 = 64}\), or making sign errors when adding terms to both sides.

These calculation mistakes lead to wrong standard form equations, which then give incorrect center coordinates. This causes them to get stuck and guess among the remaining choices.

The Bottom Line:

This problem tests whether students can systematically complete the square and correctly interpret the resulting standard form. The key insight is remembering that \(\mathrm{(x + a)^2}\) means the center coordinate is \(\mathrm{-a}\), not \(\mathrm{+a}\).