A circle in the xy-plane is defined by the equation \(3(\mathrm{x} + 1)^2 + 3(\mathrm{y} - 2)^2 = 75\). What...

1. TRANSLATE the problem information

• Given: Circle equation \(3(x + 1)^2 + 3(y - 2)^2 = 75\)
• Find: Area of this circle

2. INFER the approach needed

• To find area, we need the radius using \(\mathrm{A} = \pi\mathrm{r}^2\)
• The radius comes from the standard circle form: \((x - h)^2 + (y - k)^2 = \mathrm{r}^2\)
• Our equation isn't in standard form yet - it has a coefficient of 3

3. SIMPLIFY to convert to standard form

• Divide the entire equation by 3:
  \(3(x + 1)^2 + 3(y - 2)^2 = 75\)
  \((x + 1)^2 + (y - 2)^2 = 25\)
• Now we have standard form!

4. INFER the radius from standard form

• Comparing \((x + 1)^2 + (y - 2)^2 = 25\) to \((x - h)^2 + (y - k)^2 = \mathrm{r}^2\)
• We can see that \(\mathrm{r}^2 = 25\)

5. SIMPLIFY to find the area

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

Answer: B (\(25\pi\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students attempt to find the radius directly from the original equation without converting to standard form first.

They might try to take the square root of 75 instead of recognizing they need to divide by 3 first. This could lead them to think \(\mathrm{r}^2 = 75\), giving area = \(75\pi\), leading them to select Choice C (\(75\pi\)).

Second Most Common Error:

Poor INFER reasoning: Students correctly convert to standard form but confuse radius vs radius squared when calculating area.

They identify \(\mathrm{r}^2 = 25\) correctly but then use \(\mathrm{A} = \pi(\mathrm{r}^2)^2 = \pi(25)^2 = 625\pi\) instead of \(\mathrm{A} = \pi\mathrm{r}^2 = 25\pi\). This leads them to select Choice D (\(625\pi\)).

The Bottom Line:

This problem tests whether students can systematically convert equations to standard form before applying formulas. The key insight is that the coefficient on both squared terms must be removed first - you can't shortcut directly to the radius.