In the xy-plane, the equation x^2 + y^2 - 4x + 10y + 25 = 0 represents a circle. The...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
In the xy-plane, the equation \(\mathrm{x^2 + y^2 - 4x + 10y + 25 = 0}\) represents a circle. The point \(\mathrm{(a, -3)}\), where \(\mathrm{a}\) is a constant, lies on this circle. What is the value of the constant \(\mathrm{a}\)?
1. TRANSLATE the problem information
- Given information:
- Circle equation: \(\mathrm{x^2 + y^2 - 4x + 10y + 25 = 0}\)
- Point \(\mathrm{(a, -3)}\) lies on this circle
- Need to find: value of constant a
- What this tells us: Since the point lies on the circle, its coordinates must satisfy the circle's equation
2. SIMPLIFY by substituting the point coordinates
- Substitute \(\mathrm{x = a}\) and \(\mathrm{y = -3}\) into the equation:
\(\mathrm{a^2 + (-3)^2 - 4a + 10(-3) + 25 = 0}\)
- Expand and combine like terms:
\(\mathrm{a^2 + 9 - 4a - 30 + 25 = 0}\)
\(\mathrm{a^2 - 4a + 4 = 0}\)
3. SIMPLIFY by solving the quadratic equation
- Recognize this as a perfect square trinomial:
\(\mathrm{a^2 - 4a + 4 = (a - 2)^2}\)
- Solve: \(\mathrm{(a - 2)^2 = 0}\)
Therefore: \(\mathrm{a = 2}\)
Answer: C) 2
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skills: Students may not immediately recognize that "lies on the circle" means the point's coordinates must satisfy the circle's equation. They might try to complete the square first or attempt other unnecessary steps before realizing they simply need to substitute the given coordinates.
This leads to confusion and wasted time, potentially causing them to abandon the systematic approach and guess.
Second Most Common Error:
Poor SIMPLIFY execution: Students make arithmetic errors when substituting \(\mathrm{y = -3}\), particularly with the signs. For example, they might write \(\mathrm{10(-3)}\) as +30 instead of -30, or forget to square the -3 correctly. These errors lead to incorrect quadratic equations.
This may result in getting a different quadratic that doesn't factor to give \(\mathrm{a = 2}\), leading them to select an incorrect answer choice or become confused.
The Bottom Line:
This problem tests whether students understand the fundamental relationship between points and their curves, combined with careful algebraic manipulation. The key insight is recognizing that "lies on the circle" translates directly to "satisfies the equation."