The equation \(\mathrm{x^2 + (y + 5)^2 = 49}\) represents a circle in the xy-plane. What is the area of...

1. INFER the geometric shape

• The equation \(\mathrm{x² + (y + 5)² = 49}\) represents a circle
• I need to find the radius to calculate the area

2. TRANSLATE to standard circle form

• Standard circle equation: \(\mathrm{(x - h)² + (y - k)² = r²}\)
• Given equation: \(\mathrm{x² + (y + 5)² = 49}\)
• Rewrite as: \(\mathrm{(x - 0)² + (y - (-5))² = 49}\)

3. INFER the radius value

• Comparing to standard form: \(\mathrm{r² = 49}\)
• This means the center is \(\mathrm{(0, -5)}\) and \(\mathrm{r² = 49}\)

4. SIMPLIFY to find the radius

• Take the square root: \(\mathrm{r = \sqrt{49} = 7}\)

5. INFER the area calculation method

• Use the circle area formula: \(\mathrm{A = πr²}\)
• Substitute \(\mathrm{r = 7}\): \(\mathrm{A = π(7)² = 49π}\)

Answer: C (\(\mathrm{49π}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the equation as representing a circle, or they confuse what value represents the radius.

They might see the 49 and think it's already the radius (not \(\mathrm{r²}\)), leading them to calculate area as \(\mathrm{A = π(49)² = 2401π}\), which isn't among the choices. This leads to confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify \(\mathrm{r² = 49}\) but make an error with the area formula.

They might calculate \(\mathrm{A = π(7) = 7π}\) instead of \(\mathrm{A = π(7)² = 49π}\). This may lead them to select Choice A (\(\mathrm{7π}\)).

The Bottom Line:

This problem tests whether students can connect the standard circle equation format to extract radius information and then apply the correct area formula. The key insight is recognizing that the number on the right side of the equation (49) represents \(\mathrm{r²}\), not r itself.