In the xy-plane, a circle with radius 5 has center -{8, 6}. Which of the following is an equation of...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In the xy-plane, a circle with radius \(5\) has center \(-8, 6\). Which of the following is an equation of the circle?
\((x - 8)^2 + (y + 6)^2 = 25\)
\((x + 8)^2 + (y - 6)^2 = 25\)
\((x - 8)^2 + (y + 6)^2 = 5\)
\((x + 8)^2 + (y - 6)^2 = 5\)
1. TRANSLATE the problem information
- Given information:
- Circle has radius 5
- Center is at (-8, 6)
- Need to find the equation
2. INFER the approach
- This is asking for the equation of a circle, so I need the standard form
- Standard equation: \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
- I need to identify h, k, and r from the given information
3. TRANSLATE the given values into equation variables
- From center (-8, 6): \(\mathrm{h} = -8\) and \(\mathrm{k} = 6\)
- From radius 5: \(\mathrm{r} = 5\)
4. SIMPLIFY by substituting into the standard equation
- \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
- \((\mathrm{x} - (-8))^2 + (\mathrm{y} - 6)^2 = 5^2\)
- \((\mathrm{x} + 8)^2 + (\mathrm{y} - 6)^2 = 25\)
Answer: B. \((\mathrm{x} + 8)^2 + (\mathrm{y} - 6)^2 = 25\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students confuse which coordinate is h and which is k, or mix up the signs when dealing with negative coordinates.
For a center at (-8, 6), they might think \(\mathrm{h} = 8\) instead of \(\mathrm{h} = -8\), leading to \((\mathrm{x} - 8)^2\) instead of \((\mathrm{x} + 8)^2\). Similarly, they might think \(\mathrm{k} = -6\) instead of \(\mathrm{k} = 6\), leading to \((\mathrm{y} + 6)^2\) instead of \((\mathrm{y} - 6)^2\).
This may lead them to select Choice A (\((\mathrm{x} - 8)^2 + (\mathrm{y} + 6)^2 = 25\)) by getting both signs wrong.
Second Most Common Error:
Missing conceptual knowledge: Students forget that \(\mathrm{r}^2\) appears on the right side of the equation, not just r.
They correctly identify the center but use \(\mathrm{r} = 5\) instead of \(\mathrm{r}^2 = 25\) on the right side.
This may lead them to select Choice D (\((\mathrm{x} + 8)^2 + (\mathrm{y} - 6)^2 = 5\)) by getting the center right but the radius wrong.
The Bottom Line:
The key challenge is remembering that in \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\), the h and k values are subtracted, so a negative center coordinate becomes positive in the equation. The algebra of \(\mathrm{x} - (-8) = \mathrm{x} + 8\) is where most sign errors occur.
\((x - 8)^2 + (y + 6)^2 = 25\)
\((x + 8)^2 + (y - 6)^2 = 25\)
\((x - 8)^2 + (y + 6)^2 = 5\)
\((x + 8)^2 + (y - 6)^2 = 5\)