In the xy-plane, circle C has center at \((-4, 9)\) and area 81pi p^2. Which equation represents circle C?

1. TRANSLATE the problem information

• Given information:
  • Center: \((-4, 9)\)
  • Area: \(81\pi p^2\)
• We need to find the equation of the circle

2. INFER the solution strategy

• To write a circle equation, we need the center (given) and \(r^2\)
• We can find \(r^2\) using the area formula: \(\mathrm{A} = \pi r^2\)
• Then we'll use the standard circle equation: \((x - h)^2 + (y - k)^2 = r^2\)

3. SIMPLIFY to find \(r^2\)

• Start with: \(\mathrm{Area} = \pi r^2\)
• Substitute: \(81\pi p^2 = \pi r^2\)
• Divide both sides by \(\pi\): \(r^2 = 81p^2\)

4. TRANSLATE center coordinates to equation form

• Center \((-4, 9)\) means \(h = -4\) and \(k = 9\)
• Standard form: \((x - h)^2 + (y - k)^2 = r^2\)
• Substitute: \((x - (-4))^2 + (y - 9)^2 = 81p^2\)
• SIMPLIFY: \((x + 4)^2 + (y - 9)^2 = 81p^2\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill with signs: Students correctly find \(r^2 = 81p^2\) but make a sign error when handling the center coordinates. They see center \((-4, 9)\) and incorrectly write \((x - 4)^2\) instead of \((x + 4)^2\).

The reasoning error: They forget that \((x - h)^2\) becomes \((x - (-4))^2 = (x + 4)^2\) when \(h = -4\).

This may lead them to select Choice D: \((x - 4)^2 + (y - 9)^2 = 81p^2\)

Second Most Common Error:

Incomplete INFER reasoning about \(r^2\): Students recognize they need to use the area to find something about the radius, but they incorrectly think \(r^2\) should equal \(81\pi\) (forgetting to divide out the \(\pi\)) or just \(81\) (forgetting the \(p^2\) term).

This conceptual gap about properly isolating \(r^2\) from the area formula causes confusion and may lead them to select Choice A or B with the wrong \(r^2\) values.

The Bottom Line:

This problem requires careful attention to signs when translating center coordinates and precise algebraic manipulation to isolate \(r^2\) from the area formula. The combination of these two steps creates multiple opportunities for small but critical errors.