A circle with equation \(\mathrm{(x + 2)^2 + y^2 = 9}\) is shown in the xy-plane. A second circle, Circle...

1. TRANSLATE the first circle's equation into standard form

Given: \((x + 2)^2 + y^2 = 9\)

The standard form of a circle equation is \((x - h)^2 + (y - k)^2 = r^2\)

• Rewrite the equation to match the pattern:
  • \((x + 2)^2 = (x - (-2))^2\)
  • \(y^2 = (y - 0)^2\)
  • \(9 = 3^2\)

• So we have: \((x - (-2))^2 + (y - 0)^2 = 3^2\)

• TRANSLATE this to extract the parameters:
  • Center: \((h, k) = (-2, 0)\)
  • Radius: \(r = 3\)

2. INFER how reflection across the y-axis affects the circle

• A reflection across the y-axis transforms any point \((x, y)\) to \((-x, y)\)

• INFER what this means for the center:
  • Original center: \((-2, 0)\)
  • Reflected center: \((-(-2), 0) = (2, 0)\)

• Therefore, for Circle D: \(h = 2\) and \(k = 0\)

3. INFER the relationship between area and radius

The problem states the area of Circle D is 2 times the area of the first circle.

• Area of first circle: \(A_1 = \pi(3)^2 = 9\pi\)

• Area of Circle D: \(A_D = 2 \times 9\pi = 18\pi\)

• INFER the key insight: Since \(A = \pi r^2\), we have:
  • \(\pi r^2_D = 18\pi\)
  • \(r^2_D = 18\)

4. TRANSLATE the meaning of R in the equation

This is the critical step where students often falter!

• The equation for Circle D is given as: \((x - h)^2 + (y - k)^2 = R\)

• TRANSLATE carefully: Compare this to standard form \((x - h)^2 + (y - k)^2 = r^2\)

• Notice that R is in the position where \(r^2\) appears, NOT where r appears

• Therefore: \(R = r^2 = 18\) (not \(r = 18\))

5. SIMPLIFY to find the final answer

• We found: \(h = 2\) and \(R = 18\)

• Calculate: \(h + R = 2 + 18 = 20\)

Answer: 20

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what R represents in the equation \((x - h)^2 + (y - k)^2 = R\).

Many students assume R represents the radius r itself, not \(r^2\). Following this faulty reasoning:

• They correctly find the area of Circle D is \(18\pi\)
• They set \(\pi r^2 = 18\pi\) and get \(r^2 = 18\)
• But then they think \(r = \sqrt{18} \approx 4.24\)
• They believe \(R = \sqrt{18}\) (the radius), not \(R = 18\) (radius squared)
• Final incorrect calculation: \(h + R = 2 + \sqrt{18} \approx 2 + 4.24 = 6.24\)

This leads to confusion when they see the answer is 20, and they may second-guess their entire approach.

Second Most Common Error:

Weak INFER skill about area scaling: Students incorrectly reason that if the area doubles, the radius doubles.

Following this misconception:

• Area of first circle is \(9\pi\), so they think area doubling means something simple about the radius
• They might think: "radius = 3, so new radius = 6" (incorrectly doubling the radius)
• Then \(R = 6^2 = 36\)
• Final incorrect calculation: \(h + R = 2 + 36 = 38\)

This leads to an answer that's nowhere near correct, causing confusion and potentially guessing.

The Bottom Line:

This problem tests whether you can carefully TRANSLATE mathematical notation. The unusual form \((x - h)^2 + (y - k)^2 = R\) (using R instead of \(r^2\)) is deliberately testing if you understand what each symbol represents in the equation structure, not just pattern-matching. Always ask: "What does this symbol represent in this specific context?"