In the xy-plane, the graph of 5x^2 + 5y^2 - 30x + 10y + k = 0 is a circle...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(5\mathrm{x}^2 + 5\mathrm{y}^2 - 30\mathrm{x} + 10\mathrm{y} + \mathrm{k} = 0\)
  • This represents a circle with radius 7
• Find: value of k

2. INFER the approach needed

• To use radius information, we need the equation in standard circle form: \((\mathrm{x}-\mathrm{h})^2 + (\mathrm{y}-\mathrm{k})^2 = \mathrm{r}^2\)
• This requires completing the square for both x and y terms
• We'll need to manipulate the equation algebraically to isolate k

3. SIMPLIFY by dividing out the common factor

• Divide entire equation by 5:
  \(\mathrm{x}^2 + \mathrm{y}^2 - 6\mathrm{x} + 2\mathrm{y} + \frac{\mathrm{k}}{5} = 0\)

4. SIMPLIFY by completing the square

• For x terms: \(\mathrm{x}^2 - 6\mathrm{x}\)
  • Take half the coefficient of x: \(-6/2 = -3\)
  • Square it: \((-3)^2 = 9\)
  • So: \(\mathrm{x}^2 - 6\mathrm{x} = (\mathrm{x} - 3)^2 - 9\)
• For y terms: \(\mathrm{y}^2 + 2\mathrm{y}\)
  • Take half the coefficient of y: \(2/2 = 1\)
  • Square it: \(1^2 = 1\)
  • So: \(\mathrm{y}^2 + 2\mathrm{y} = (\mathrm{y} + 1)^2 - 1\)

5. SIMPLIFY by substituting and rearranging

• Replace completed square forms:
  \((\mathrm{x} - 3)^2 - 9 + (\mathrm{y} + 1)^2 - 1 + \frac{\mathrm{k}}{5} = 0\)
• Combine constants:
  \((\mathrm{x} - 3)^2 + (\mathrm{y} + 1)^2 - 10 + \frac{\mathrm{k}}{5} = 0\)
• Rearrange to standard form:
  \((\mathrm{x} - 3)^2 + (\mathrm{y} + 1)^2 = 10 - \frac{\mathrm{k}}{5}\)

6. INFER the radius relationship and solve

• Since radius = 7, then \(\mathrm{r}^2 = 49\)
• Set our equation equal to 49:
  \(49 = 10 - \frac{\mathrm{k}}{5}\)
• Solve for k:
  \(\frac{\mathrm{k}}{5} = 10 - 49 = -39\)
  \(\mathrm{k} = -195\)

Answer: C (-195)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic errors while completing the square, particularly with signs and constants.

For example, they might incorrectly complete \(\mathrm{x}^2 - 6\mathrm{x}\) as \((\mathrm{x} - 3)^2 - 6\) instead of \((\mathrm{x} - 3)^2 - 9\), or make sign errors when combining the constant terms (-9, -1, and k/5). These algebraic mistakes lead to an incorrect final equation and wrong value of k.

This may lead them to select Choice A (-205) or Choice B (-200).

Second Most Common Error:

Missing conceptual knowledge: Students don't remember that completing the square requires subtracting the squared constant.

They might write \(\mathrm{x}^2 - 6\mathrm{x} = (\mathrm{x} - 3)^2\) without subtracting 9, leading to an equation that doesn't properly represent the original circle. This fundamental gap in completing the square technique makes the entire solution incorrect.

This causes them to get stuck and guess among the answer choices.

The Bottom Line:

This problem requires solid algebra skills in completing the square combined with understanding circle equations. Students who rush through the algebraic steps or lack confidence in completing the square will struggle to reach the correct answer systematically.