Question:Circle C in the xy-plane has the equation \((\mathrm{x} - 1)^2 + (\mathrm{y} + 2)^2 = 9\). Circle D has...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
Circle C in the xy-plane has the equation \((\mathrm{x} - 1)^2 + (\mathrm{y} + 2)^2 = 9\). Circle D has the same center as circle C. The radius of circle D is three times the radius of circle C. What is the diameter of circle D?
Enter your answer as an integer.
1. TRANSLATE the circle equation to identify key information
- Given information:
- Circle C: \((x - 1)^2 + (y + 2)^2 = 9\)
- Circle D has the same center as Circle C
- Circle D's radius is three times Circle C's radius
- What this tells us: We need to extract the center and radius from the standard form equation.
2. INFER what the standard form reveals
- The equation \((x - 1)^2 + (y + 2)^2 = 9\) matches the standard form \((x - h)^2 + (y - k)^2 = r^2\)
- This means: \(\mathrm{center} = (1, -2)\) and \(r^2 = 9\)
3. SIMPLIFY to find Circle C's radius
- Since \(r^2 = 9\), we have \(r = \sqrt{9} = 3\)
- Circle C has \(\mathrm{radius} = 3\)
4. INFER Circle D's properties
- Circle D has the same center: \((1, -2)\)
- Circle D's radius = \(3 \times \mathrm{(Circle\ C's\ radius)} = 3 \times 3 = 9\)
5. TRANSLATE from radius to diameter
- Diameter = \(2 \times \mathrm{radius} = 2 \times 9 = 18\)
Answer: 18
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may not recognize the standard form \((x - h)^2 + (y + 2)^2 = r^2\) or incorrectly identify the radius.
Some students might think \(r = 9\) instead of \(r^2 = 9\), leading them to calculate Circle D's radius as \(9 \times 3 = 27\), and diameter as 54. This leads to confusion since 54 isn't among typical answer choices, causing them to guess.
Second Most Common Error:
Incomplete INFER reasoning: Students correctly find Circle C's radius as 3 and Circle D's radius as 9, but forget the final step of converting radius to diameter.
This causes them to answer 9 instead of 18, stopping one step short of the complete solution.
The Bottom Line:
This problem tests whether students can work systematically through multiple connected steps: extracting information from standard form, applying scaling relationships, and converting between radius and diameter. Success requires maintaining focus through the entire sequence rather than stopping at any intermediate result.