In the xy-plane, a point \(\mathrm{(x, y)}\) lies on the circle x^2 + y^2 = 65. The point also lies...

1. TRANSLATE the problem information

• Given information:
  • Point (x, y) lies on circle: \(\mathrm{x^2 + y^2 = 65}\)
  • Same point lies on line: \(\mathrm{y = 8}\)
  • Need to find possible x-coordinate

• What this tells us: Since the point satisfies both equations simultaneously, we can substitute one equation into the other.

2. INFER the solution strategy

• Since we know \(\mathrm{y = 8}\), we can substitute this directly into the circle equation
• This will give us a simple equation in terms of x only

3. SIMPLIFY through substitution and algebra

• Substitute \(\mathrm{y = 8}\) into \(\mathrm{x^2 + y^2 = 65}\):
  \(\mathrm{x^2 + 8^2 = 65}\)
  \(\mathrm{x^2 + 64 = 65}\)
  \(\mathrm{x^2 = 1}\)

4. CONSIDER ALL CASES for the square root

• Taking square root of both sides: \(\mathrm{x = \pm 1}\)
• So x can be either 1 or -1

5. APPLY CONSTRAINTS from answer choices

• Looking at the given options: (A) -8, (B) -1, (C) 0, (D) 8
• Only -1 appears among the choices

Answer: B (-1)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly substitute \(\mathrm{y = 8}\) but make arithmetic errors when solving \(\mathrm{x^2 + 64 = 65}\), perhaps getting \(\mathrm{x^2 = 129}\) instead of \(\mathrm{x^2 = 1}\).

This leads to confusion when trying to find square roots and may cause them to abandon systematic solving and guess randomly.

Second Most Common Error:

Poor CONSIDER ALL CASES reasoning: Students find \(\mathrm{x^2 = 1}\) and correctly get \(\mathrm{x = \pm 1}\), but then assume they need to find both values or don't realize they should check which appears in the answer choices.

This may lead them to get stuck since they see two possible answers but only one choice matches.

The Bottom Line:

This problem tests whether students can systematically handle the intersection of two curves through substitution. The key insight is that when a point lies on multiple curves, its coordinates must satisfy all equations simultaneously - leading to a simple substitution strategy.