Circle A has equation \(\mathrm{x^2 + (y - 3)^2 = 16}\). Circle B is obtained by reflecting circle A across...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
Circle A has equation \(\mathrm{x^2 + (y - 3)^2 = 16}\). Circle B is obtained by reflecting circle A across the x-axis. Which of the following equations represents circle B?
\(\mathrm{x^2 + (y - 3)^2 = 16}\)
\(\mathrm{x^2 + (y + 3)^2 = 16}\)
\(\mathrm{(x - 3)^2 + y^2 = 16}\)
\(\mathrm{x^2 + (-y + 3)^2 = 16}\)
1. TRANSLATE the given information
- Given: Circle A has equation \(\mathrm{x^2 + (y - 3)^2 = 16}\)
- Need: Equation of Circle B after reflecting Circle A across the x-axis
2. INFER the center and radius of Circle A
- From standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\):
- Center: \(\mathrm{(0, 3)}\)
- Radius: \(\mathrm{\sqrt{16} = 4}\)
3. TRANSLATE the reflection rule
- Reflecting across the x-axis means:
- x-coordinate stays the same
- y-coordinate changes sign
- So point \(\mathrm{(0, 3)}\) becomes \(\mathrm{(0, -3)}\)
4. INFER what changes and what stays the same
- Circle B will have:
- Same x-coordinate: 0
- New y-coordinate: -3
- Same radius: 4
5. SIMPLIFY to write Circle B's equation
- Center \(\mathrm{(0, -3)}\) and radius 4 gives us:
- \(\mathrm{x^2 + (y - (-3))^2 = 16}\)
- \(\mathrm{x^2 + (y + 3)^2 = 16}\)
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students incorrectly interpret "reflecting across the x-axis" and think both coordinates change, or they get confused about which coordinate changes sign.
They might think the x-coordinate changes instead, leading them to move the center from \(\mathrm{(0, 3)}\) to \(\mathrm{(-3, 0)}\), which would give equation \(\mathrm{(x + 3)^2 + (y - 3)^2 = 16}\). Since this isn't an option, they get confused and may select Choice C (\(\mathrm{(x - 3)^2 + y^2 = 16}\)) thinking it's somehow related to moving coordinates around.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify that the center becomes \(\mathrm{(0, -3)}\) but make an algebra error when writing the equation.
They might write \(\mathrm{x^2 + (y - (-3))^2 = 16}\) but then incorrectly simplify this as \(\mathrm{x^2 + (-y + 3)^2 = 16}\), leading them to select Choice D.
The Bottom Line:
This problem tests whether students truly understand coordinate transformations and can systematically apply the reflection rule while maintaining careful algebra throughout.
\(\mathrm{x^2 + (y - 3)^2 = 16}\)
\(\mathrm{x^2 + (y + 3)^2 = 16}\)
\(\mathrm{(x - 3)^2 + y^2 = 16}\)
\(\mathrm{x^2 + (-y + 3)^2 = 16}\)