\((\mathrm{x} - 6)^2 + (\mathrm{y} - 3)^2 = 81\) The graph of the given equation in the xy-plane is a...

1. INFER the equation type

• Given: \((\mathrm{x} - 6)^2 + (\mathrm{y} - 3)^2 = 81\)
• This matches the standard form of a circle equation: \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• Strategy: Extract the center and radius from this form

2. INFER the components

• Comparing \((\mathrm{x} - 6)^2 + (\mathrm{y} - 3)^2 = 81\) to \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\):
  • \(\mathrm{h} = 6, \mathrm{k} = 3\) (so center is \((6, 3)\))
  • \(\mathrm{r}^2 = 81\)

3. SIMPLIFY to find the radius

• We have \(\mathrm{r}^2 = 81\)
• Take the square root: \(\mathrm{r} = \sqrt{81} = 9\)
• Since radius must be positive, \(\mathrm{r} = 9\)

Answer: C. 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Students don't recognize the standard form of a circle equation or confuse what each component represents.

Without knowing that \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\) is the standard form, they might try to expand the equation or use unrelated formulas. This leads to confusion and guessing.

Second Most Common Error:

Weak SIMPLIFY skill: Students correctly identify that \(\mathrm{r}^2 = 81\) but select 81 as the radius instead of taking the square root.

They stop one step short of the complete solution, thinking the constant term directly gives the radius. This may lead them to select Choice D (81).

The Bottom Line:

Success requires both recognizing circle equations and understanding that the constant term gives \(\mathrm{r}^2\), not r directly.