What is the center of the circle in the xy-plane defined by the equation \((\mathrm{x} - 1)^2 + (\mathrm{y} +...

1. TRANSLATE the given equation to standard form pattern

• Given equation: \((x - 1)^2 + (y + 7)^2 = 1\)
• Standard form: \((x - h)^2 + (y - k)^2 = r^2\)
• We need to identify h, k, and r values

2. TRANSLATE the x-term to find h

• \((x - 1)^2\) matches \((x - h)^2\)
• Therefore: \(h = 1\)

3. TRANSLATE the y-term to find k

• \((y + 7)^2\) needs to match \((y - k)^2\)
• Rewrite: \((y + 7)^2 = (y - (-7))^2\)
• Therefore: \(k = -7\)

4. INFER the center coordinates

• Since \((h, k)\) represents the center in standard form
• Center = \((1, -7)\)

Answer: C. (1, -7)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students incorrectly handle the signs when identifying k from \((y + 7)^2\).

They see \((y + 7)^2\) and think "\(k = +7\)" instead of recognizing that \((y + 7)^2 = (y - (-7))^2\), so \(k = -7\). This sign error is especially common because students rush through the pattern matching without carefully considering what the standard form actually requires.

This may lead them to select Choice D (1, 7).

The Bottom Line:

The key challenge is correctly handling the sign conversion in the y-term. Students must recognize that \((y + 7)^2\) means we're subtracting negative 7, not positive 7, from y in the standard form.