\((\mathrm{x} - 4)^2 + (\mathrm{y} + 2)^2 = 25\) The graph of the given equation in the xy-plane is a...

1. INFER the equation type

• The equation \((\mathrm{x} - 4)^2 + (\mathrm{y} + 2)^2 = 25\) matches the standard form of a circle
• Standard form: \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\) where \((\mathrm{h},\mathrm{k})\) is center and \(\mathrm{r}\) is radius
• This tells us we have a circle and can find its radius

2. SIMPLIFY to find the radius

• Comparing \((\mathrm{x} - 4)^2 + (\mathrm{y} + 2)^2 = 25\) with \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• We see that \(\mathrm{r}^2 = 25\)
• Taking the square root: \(\mathrm{r} = \sqrt{25} = 5\)

3. INFER what we need for area

• To find area of circle, we use \(\mathrm{A} = \pi\mathrm{r}^2\)
• We have \(\mathrm{r} = 5\), so we can substitute

4. SIMPLIFY the area calculation

• \(\mathrm{A} = \pi\mathrm{r}^2\)
• \(\mathrm{A} = \pi(5)^2\)
• \(\mathrm{A} = \pi(25)\)
• \(\mathrm{A} = 25\pi\)

Answer: D) \(25\pi\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about area formula: Students may see \(\mathrm{r}^2 = 25\) and think the area is just 25, forgetting that circle area requires multiplying by \(\pi\).

This reasoning makes them think "the area must be 25" and leads them to select Choice C (25).

Second Most Common Error:

Weak INFER skill: Students recognize it's a circle and find \(\mathrm{r} = 5\) correctly, but then confuse radius with area, thinking "if radius is 5, then area is 5."

This may lead them to select Choice A (5) or if they remember \(\pi\) is involved somehow, Choice B (\(5\pi\)).

The Bottom Line:

This problem requires students to distinguish between the radius value (5) and what goes into the area formula (\(\pi\mathrm{r}^2 = \pi \cdot 25 = 25\pi\)). Many students correctly identify the radius but then lose track of the area formula structure.