In the xy-plane, the graph of the equation \((\mathrm{x} - 3)^2 + (\mathrm{y} - 5)^2 = 9\) is a circle....
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In the xy-plane, the graph of the equation \((\mathrm{x} - 3)^2 + (\mathrm{y} - 5)^2 = 9\) is a circle. The point \((6, \mathrm{c})\), where \(\mathrm{c}\) is a constant, lies on this circle. What is the value of \(\mathrm{c}\)?
1. TRANSLATE the problem information
- Given information:
- Circle equation: \((\mathrm{x} - 3)^2 + (\mathrm{y} - 5)^2 = 9\)
- Point \((6, \mathrm{c})\) lies on this circle
- Need to find the value of c
- What this tells us: Since the point lies ON the circle, its coordinates must satisfy the circle equation when substituted.
2. TRANSLATE the mathematical requirement
- If point \((6, \mathrm{c})\) lies on the circle, then substituting \(\mathrm{x} = 6\) and \(\mathrm{y} = \mathrm{c}\) into the equation must make it true
- This gives us: \((6 - 3)^2 + (\mathrm{c} - 5)^2 = 9\)
3. SIMPLIFY the equation step by step
- Calculate \((6 - 3)^2\):
\((6 - 3)^2 = 3^2 = 9\)
- Substitute this result:
\(9 + (\mathrm{c} - 5)^2 = 9\)
- Subtract 9 from both sides:
\((\mathrm{c} - 5)^2 = 0\)
4. SIMPLIFY to find c
- Take the square root of both sides:
\(\mathrm{c} - 5 = 0\)
- Solve for c:
\(\mathrm{c} = 5\)
Answer: 5
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may not understand what "lies on the circle" means mathematically. They might try to work with the circle's center or radius instead of recognizing that they need to substitute the point's coordinates into the equation.
This leads to confusion about the problem setup and may cause them to attempt incorrect approaches or guess randomly.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up \((6 - 3)^2 + (\mathrm{c} - 5)^2 = 9\) but make arithmetic errors, such as:
- Calculating \((6 - 3)^2\) incorrectly
- Making sign errors when rearranging the equation
- Incorrectly solving \((\mathrm{c} - 5)^2 = 0\)
These computational mistakes lead to incorrect values of c.
The Bottom Line:
This problem tests whether students understand the fundamental relationship between a point and the curve it lies on - the coordinates must satisfy the equation. Once this connection is made, the algebra is straightforward.