In the xy-plane, the circle C has equation \((\mathrm{x} − \mathrm{a})² + (\mathrm{y} + 6)² = 49\), where a is...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In the xy-plane, the circle C has equation \((\mathrm{x} − \mathrm{a})² + (\mathrm{y} + 6)² = 49\), where \(\mathrm{a}\) is a real number. Circle C is translated \(2\) units to the left and \(5\) units downward to form circle D. Which of the following gives the center of circle D and its radius?
1. TRANSLATE the circle equation to identify original properties
- Given equation: \(\mathrm{(x - a)^2 + (y + 6)^2 = 49}\)
- This is in standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\)
- INFER the center and radius:
- Center: \(\mathrm{(a, -6)}\) [since \(\mathrm{(y + 6)^2}\) means \(\mathrm{k = -6}\)]
- Radius: \(\mathrm{\sqrt{49} = 7}\)
2. TRANSLATE the translation description into coordinate changes
- "2 units to the left" means x-coordinate decreases by 2
- "5 units downward" means y-coordinate decreases by 5
- INFER that translation doesn't change the radius
3. Apply the translation to find circle D's center
- Original center: \(\mathrm{(a, -6)}\)
- After moving 2 units left: \(\mathrm{a \rightarrow a - 2}\)
- After moving 5 units down: \(\mathrm{-6 \rightarrow -6 - 5 = -11}\)
- New center: \(\mathrm{(a - 2, -11)}\)
- Radius remains: 7
Answer: A. The center is at (a - 2, -11) and the radius is 7.
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret the direction of translation, thinking "2 units to the left" means adding 2 to the x-coordinate instead of subtracting 2, or thinking "5 units downward" means adding 5 to the y-coordinate instead of subtracting 5.
This incorrect reasoning might lead them to calculate the new center as \(\mathrm{(a + 2, -6 + 5) = (a + 2, -1)}\), which doesn't match any answer choice exactly but might cause them to select Choice B (a + 2, 11) if they also make sign errors.
Second Most Common Error:
Missing conceptual knowledge: Students don't recognize the standard form of the circle equation and incorrectly identify the original center, perhaps thinking the center is \(\mathrm{(-a, 6)}\) instead of \(\mathrm{(a, -6)}\).
This leads to applying the translation to the wrong starting point, resulting in a center like \(\mathrm{(-a - 2, 6 - 5) = (-a - 2, 1)}\), which might make them select Choice D if they also make additional errors.
The Bottom Line:
Translation problems require careful attention to coordinate directions and signs. The key insight is that "left" and "down" both involve subtraction, while the radius never changes during translation.