In the xy-plane, the equation x^2 + y^2 - 6x - 8y + k = 0 represents a circle. The...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In the xy-plane, the equation \(\mathrm{x^2 + y^2 - 6x - 8y + k = 0}\) represents a circle. The point \(\mathrm{(3, 1)}\) lies on this circle. What is the value of k?
Express your answer as an integer.
1. TRANSLATE the problem information
- Given information:
- Circle equation: \(\mathrm{x^2 + y^2 - 6x - 8y + k = 0}\)
- Point (3, 1) lies on this circle
- What this tells us: Since the point lies on the circle, its coordinates must satisfy the circle's equation
2. TRANSLATE the mathematical approach
- If point (3, 1) is on the circle, then \(\mathrm{x = 3}\) and \(\mathrm{y = 1}\) must make the equation true
- Strategy: Substitute these values and solve for k
3. SIMPLIFY by substituting the coordinates
- Replace x with 3 and y with 1 in the equation:
\(\mathrm{(3)^2 + (1)^2 - 6(3) - 8(1) + k = 0}\)
4. SIMPLIFY the arithmetic step by step
- Calculate each term:
- \(\mathrm{(3)^2 = 9}\)
- \(\mathrm{(1)^2 = 1}\)
- \(\mathrm{-6(3) = -18}\)
- \(\mathrm{-8(1) = -8}\)
- Substitute: \(\mathrm{9 + 1 - 18 - 8 + k = 0}\)
5. SIMPLIFY to solve for k
- Combine like terms: \(\mathrm{10 - 26 + k = 0}\)
- Simplify: \(\mathrm{-16 + k = 0}\)
- Therefore: \(\mathrm{k = 16}\)
Answer: 16
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE reasoning: Not understanding that "lies on the circle" means the point's coordinates satisfy the equation.
Students might think they need to complete the square to find the center and radius first, or they might not know how to use the given point. This leads to confusion and guessing rather than the direct substitution approach.
Second Most Common Error:
Poor SIMPLIFY execution: Making arithmetic errors, especially with negative signs.
Students might calculate correctly up to \(\mathrm{9 + 1 - 18 - 8 + k = 0}\), but then make errors like:
- Getting \(\mathrm{10 - 10 + k = 0}\) (incorrectly combining -18 and -8)
- Sign errors when isolating k
This could lead them to incorrect values like \(\mathrm{k = 0}\) or other wrong answers.
The Bottom Line:
This problem tests whether students understand the fundamental relationship between points and equations - that coordinates of points on a curve must satisfy the curve's equation. The computation is straightforward once this connection is made.