In the xy-plane, the equation of circle P is \((x - 8)^2 + (y + 1)^2 = 5\). Circle Q...

1. TRANSLATE the given information

• Circle P: \((\mathrm{x} - 8)^2 + (\mathrm{y} + 1)^2 = 5\)
• Circle Q has same center as P
• Area of Q = 5 × Area of P
• Circle Q: \((\mathrm{x} - 8)^2 + (\mathrm{y} + 1)^2 = \mathrm{c}\)
• Find: value of c

2. INFER what the equation tells us about circle P

• From standard form \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\):
  • Center of P: (8, -1)
  • Radius squared: \(\mathrm{r}_\mathrm{P}^2 = 5\)
  • Area of P: \(\mathrm{A}_\mathrm{P} = \pi\mathrm{r}_\mathrm{P}^2 = \pi(5) = 5\pi\)

3. TRANSLATE the area relationship for circle Q

• Area of Q = 5 × Area of P
• \(\mathrm{A}_\mathrm{Q} = 5 \times 5\pi = 25\pi\)

4. INFER the radius of circle Q

• Since \(\mathrm{A}_\mathrm{Q} = \pi(\mathrm{r}_\mathrm{Q})^2\), we have:
• \(\pi(\mathrm{r}_\mathrm{Q})^2 = 25\pi\)

5. SIMPLIFY to find radius squared of circle Q

• Divide both sides by \(\pi\): \((\mathrm{r}_\mathrm{Q})^2 = 25\)
• Therefore: \(\mathrm{r}_\mathrm{Q}^2 = 25\)

6. INFER the value of c

• Circle Q equation: \((\mathrm{x} - 8)^2 + (\mathrm{y} + 1)^2 = \mathrm{c}\)
• In standard form, c equals radius squared
• Therefore: \(\mathrm{c} = \mathrm{r}_\mathrm{Q}^2 = 25\)

Answer: 25

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may misinterpret "5 times the area" as meaning the radius is 5 times larger, leading them to think \(\mathrm{r}_\mathrm{Q} = 5\mathrm{r}_\mathrm{P} = 5\sqrt{5}\).

From here, they calculate \((\mathrm{r}_\mathrm{Q})^2 = (5\sqrt{5})^2 = 25(5) = 125\), giving \(\mathrm{c} = 125\).

This leads to confusion since 125 isn't typically among the answer choices, causing students to guess or abandon their systematic approach.

Second Most Common Error:

Missing conceptual knowledge about standard form: Students may not recognize that in \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\), the constant term equals \(\mathrm{r}^2\), not r.

They might think c represents the radius itself rather than radius squared, leading them to conclude \(\mathrm{c} = 5\) (since they correctly find \(\mathrm{r}_\mathrm{Q} = 5\) from the area relationship).

The Bottom Line:

This problem tests whether students can distinguish between linear scaling (like radius) versus quadratic scaling (like area), and whether they understand the standard form of circle equations. The key insight is that areas scale by the square of the radius scaling factor.