A circle in the xy-plane has its center at \((-5, 2)\) and has a radius of 9. An equation of...

1. TRANSLATE the given information into mathematical form

• Given information:
  • Center: \((-5, 2)\)
  • Radius: \(9\)
  • Target form: \(\mathrm{x^2 + y^2 + ax + by + c = 0}\)

• What this tells us: We need to start with the standard circle equation and transform it.

2. INFER the approach

• Since we know center and radius, start with standard form: \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• Then expand and rearrange to match the target form
• The coefficient \(\mathrm{c}\) will be our final answer

3. TRANSLATE into standard circle equation

• Substitute \(\mathrm{h = -5, k = 2, r = 9}\):
  \((\mathrm{x} - (-5))^2 + (\mathrm{y} - 2)^2 = 9^2\)
  \((\mathrm{x} + 5)^2 + (\mathrm{y} - 2)^2 = 81\)

4. SIMPLIFY by expanding perfect squares

• \((\mathrm{x} + 5)^2 = \mathrm{x}^2 + 10\mathrm{x} + 25\)
• \((\mathrm{y} - 2)^2 = \mathrm{y}^2 - 4\mathrm{y} + 4\)

• Equation becomes:
  \(\mathrm{x}^2 + 10\mathrm{x} + 25 + \mathrm{y}^2 - 4\mathrm{y} + 4 = 81\)

5. SIMPLIFY by collecting and rearranging terms

• Combine like terms on left side:
  \(\mathrm{x}^2 + \mathrm{y}^2 + 10\mathrm{x} - 4\mathrm{y} + 29 = 81\)

• Move 81 to left side:
  \(\mathrm{x}^2 + \mathrm{y}^2 + 10\mathrm{x} - 4\mathrm{y} + 29 - 81 = 0\)
  \(\mathrm{x}^2 + \mathrm{y}^2 + 10\mathrm{x} - 4\mathrm{y} - 52 = 0\)

• In the form \(\mathrm{x^2 + y^2 + ax + by + c = 0}\):
  \(\mathrm{c = -52}\)

Answer: -52

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when substituting \(\mathrm{h = -5}\) into \((\mathrm{x} - \mathrm{h})^2\).

Students often write \((\mathrm{x} - (-5))^2\) correctly but then expand it as \((\mathrm{x} - 5)^2\) instead of \((\mathrm{x} + 5)^2\), leading to \(\mathrm{x}^2 - 10\mathrm{x} + 25\) instead of \(\mathrm{x}^2 + 10\mathrm{x} + 25\). This error cascades through the remaining calculations, ultimately giving \(\mathrm{c = -108}\) instead of \(-52\).

Second Most Common Error:

Poor SIMPLIFY reasoning: Arithmetic mistakes when combining the constant terms.

Students correctly expand both perfect squares but make errors when calculating \(25 + 4 - 81\). Some calculate this as \(25 + 4 + 81 = 110\) (forgetting the subtraction) or simply miscalculate the arithmetic, leading to various incorrect values for \(\mathrm{c}\).

The Bottom Line:

This problem tests your ability to bridge geometric concepts with algebraic manipulation. The key insight is recognizing that any circle can be written in multiple equivalent forms, and careful algebraic work is essential to avoid sign and arithmetic errors during the conversion process.