Circle A in the xy-plane has the equation \((\mathrm{x} + 5)^2 + (\mathrm{y} - 5)^2 = 4\). Circle B has...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
Circle A in the xy-plane has the equation \((\mathrm{x} + 5)^2 + (\mathrm{y} - 5)^2 = 4\). Circle B has the same center as circle A. The radius of circle B is two times the radius of circle A. The equation defining circle B in the xy-plane is \((\mathrm{x} + 5)^2 + (\mathrm{y} - 5)^2 = \mathrm{k}\), where \(\mathrm{k}\) is a constant. What is the value of \(\mathrm{k}\)?
1. TRANSLATE Circle A's equation to find its properties
- Given equation: \((x + 5)^2 + (y - 5)^2 = 4\)
- TRANSLATE this to standard form \((x - h)^2 + (y - k)^2 = r^2\):
- Rewrite as: \((x - (-5))^2 + (y - 5)^2 = 2^2\)
- Center (h, k): \((-5, 5)\)
- Radius r: 2
2. TRANSLATE the relationship between circles A and B
- Given information:
- Circle B has same center as Circle A
- Radius of B = 2 × radius of A
- This tells us:
- Center of B: (-5, 5) [same as A]
- Radius of B: \(2 \times 2 = 4\)
3. INFER what form Circle B's equation takes
- Since B has center (-5, 5) and radius 4, its standard form is:
\((x - (-5))^2 + (y - 5)^2 = 4^2\)
4. SIMPLIFY to match the given format
- \((x - (-5))^2 + (y - 5)^2 = 4^2\)
- \((x + 5)^2 + (y - 5)^2 = 16\)
- Comparing with \((x + 5)^2 + (y - 5)^2 = k\):
\(k = 16\)
Answer: 16
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students correctly find that Circle B has radius 4, but then set \(k = 4\) instead of \(k = 16\).
The reasoning error is thinking "radius is 4, so k = 4" without recognizing that k represents r², not r. In the standard form \((x - h)^2 + (y - k)^2 = r^2\), the right side is the radius squared, not just the radius.
This leads them to incorrectly conclude \(k = 4\).
Second Most Common Error:
Inadequate INFER skill: Students see "radius of B is two times radius of A" and double the 4 from Circle A's equation, thinking k should be \(2 \times 4 = 8\).
This happens when they don't first TRANSLATE Circle A's equation properly to realize that the 4 represents r², not r. They treat the 4 as if it's the radius, then double it.
This leads them to incorrectly conclude \(k = 8\).
The Bottom Line:
The key challenge is recognizing that in \((x - h)^2 + (y - k)^2 = r^2\), the constant on the right side is r² (radius squared), not r (radius). Students must extract the actual radius first, apply the relationship, then square the new radius.