Question:In the xy-plane, a circle is defined by the equation x^2 + y^2 - 4x + 10y + 20 =...

1. INFER the solution strategy

• Given: Circle equation \(\mathrm{x^2 + y^2 - 4x + 10y + 20 = 0}\)
• Need: Center coordinates \(\mathrm{(h, k)}\)
• Strategy: Convert to standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\) by completing the square

2. SIMPLIFY by rearranging and grouping terms

• Move constant to right side: \(\mathrm{x^2 + y^2 - 4x + 10y = -20}\)
• Group x and y terms: \(\mathrm{(x^2 - 4x) + (y^2 + 10y) = -20}\)

3. SIMPLIFY by completing the square for x

• For \(\mathrm{x^2 - 4x}\): take half of coefficient (-4), then square it
• \(\mathrm{(-4/2)^2 = (-2)^2 = 4}\)
• Add and subtract: \(\mathrm{x^2 - 4x + 4 - 4 = (x - 2)^2 - 4}\)

4. SIMPLIFY by completing the square for y

• For \(\mathrm{y^2 + 10y}\): take half of coefficient (10), then square it
• \(\mathrm{(10/2)^2 = 5^2 = 25}\)
• Add and subtract: \(\mathrm{y^2 + 10y + 25 - 25 = (y + 5)^2 - 25}\)

5. SIMPLIFY the complete equation

• Substitute completed squares: \(\mathrm{(x - 2)^2 - 4 + (y + 5)^2 - 25 = -20}\)
• Combine constants: \(\mathrm{(x - 2)^2 + (y + 5)^2 = -20 + 4 + 25 = 9}\)

6. INFER the center from standard form

• Standard form: \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\)
• Our form: \(\mathrm{(x - 2)^2 + (y + 5)^2 = 9}\)
• Reading coordinates: \(\mathrm{h = 2, k = -5}\) (note: \(\mathrm{y + 5}\) means \(\mathrm{y - (-5)}\))
• Center: \(\mathrm{(2, -5)}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when completing the square or reading the final coordinates. The most frequent mistake is with the y-coordinate - seeing \(\mathrm{(y + 5)^2}\) and thinking the center has y-coordinate +5 instead of -5.

This leads them to select Choice B \(\mathrm{((-2, 5))}\) or Choice D \(\mathrm{((2, 5))}\) instead of the correct \(\mathrm{(2, -5)}\).

Second Most Common Error:

Incomplete INFER strategy: Students attempt to complete the square but make arithmetic errors in the process, such as incorrectly calculating \(\mathrm{(b/2)^2}\) or making errors when combining constants on the right side.

This leads to confusion and incorrect center coordinates, causing them to guess among the answer choices.

The Bottom Line:

This problem tests whether students can execute the multi-step completing the square process accurately while carefully tracking signs throughout. The key insight is that \(\mathrm{(y + 5)^2}\) means the center's y-coordinate is -5, not +5.