A circle in the xy-plane has the equation \((\mathrm{x} - 13)^2 + (\mathrm{y} - \mathrm{k})^2 = 64\). Which of the...

1. INFER the equation pattern

• Given equation: \((\mathrm{x} - 13)^2 + (\mathrm{y} - \mathrm{k})^2 = 64\)
• This matches the standard circle form: \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• Key insight: The numbers in the parentheses tell us the center coordinates, and the right side tells us \(\mathrm{r}^2\)

2. TRANSLATE the equation components

• From \((\mathrm{x} - 13)^2\): the center's x-coordinate is \(\mathrm{h} = 13\)
• From \((\mathrm{y} - \mathrm{k})^2\): the center's y-coordinate is \(\mathrm{k}\) (already given as a variable)
• From \(= 64\): we have \(\mathrm{r}^2 = 64\)

3. SIMPLIFY to find the radius

• \(\mathrm{r}^2 = 64\)
• \(\mathrm{r} = \sqrt{64} = 8\)

4. INFER the final answer

• Center coordinates: \((\mathrm{h}, \mathrm{k}) = (13, \mathrm{k})\)
• Radius: \(\mathrm{r} = 8\)

Answer: A. The center is at \((13, \mathrm{k})\) and the radius is 8.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students confuse the coordinate order and think the center should be \((\mathrm{k}, 13)\) instead of \((13, \mathrm{k})\).

They see the equation \((\mathrm{x} - 13)^2 + (\mathrm{y} - \mathrm{k})^2 = 64\) and incorrectly match the variables, thinking that since k appears with y, the center must be \((\mathrm{k}, 13)\). This fundamental misunderstanding of how the standard form works leads them to flip the coordinates.

This may lead them to select Choice B (\((\mathrm{k}, 13)\) and radius 8).

Second Most Common Error:

Poor SIMPLIFY execution: Students mistake \(\mathrm{r}^2 = 64\) for the actual radius, forgetting to take the square root.

They correctly identify the center as \((13, \mathrm{k})\) but then state the radius as 64 instead of recognizing that they need to find \(\sqrt{64} = 8\). This happens when students rush through the final step or don't fully understand that the right side of the equation represents \(\mathrm{r}^2\), not \(\mathrm{r}\) directly.

This may lead them to select Choice D (\((13, \mathrm{k})\) and radius 64).

The Bottom Line:

This problem tests whether students truly understand the structure of the standard circle equation. Success requires careful attention to coordinate order and remembering that the equation gives \(\mathrm{r}^2\), not \(\mathrm{r}\) directly.