The equation \((\mathrm{x} + 6)^2 + (\mathrm{y} + 3)^2 = 121\) defines a circle in the xy-plane. What is the...

1. TRANSLATE the equation into recognizable form

• Given equation: \((x + 6)^2 + (y + 3)^2 = 121\)
• This matches the standard form of a circle: \((x - h)^2 + (y - k)^2 = r^2\)
  • \((h, k)\) = center coordinates
  • \(r\) = radius

2. TRANSLATE to identify the radius term

• Comparing \((x + 6)^2 + (y + 3)^2 = 121\) to \((x - h)^2 + (y - k)^2 = r^2\)
• The right side gives us: \(r^2 = 121\)

3. SIMPLIFY to find the radius

• Take the square root: \(r = \sqrt{121} = 11\)
• Since radius represents a distance, we take the positive value

Answer: 11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not recognizing the standard circle equation format

Students might see \((x + 6)^2 + (y + 3)^2 = 121\) and not immediately connect it to the circle formula \((x - h)^2 + (y - k)^2 = r^2\). They may get confused by the plus signs in front of the 6 and 3, thinking this doesn't match the standard form. This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Calculation error in SIMPLIFY: Incorrectly calculating \(\sqrt{121}\)

Some students might make arithmetic errors when finding \(\sqrt{121}\), perhaps confusing it with other perfect squares like \(\sqrt{144} = 12\) or \(\sqrt{100} = 10\). This could lead them to select an incorrect numerical answer.

The Bottom Line:

Success on this problem hinges on pattern recognition - seeing that the equation matches the standard circle form and knowing that \(r^2\) equals the constant term on the right side.