In the coordinate plane, point P lies on a circle centered at the origin with radius 17 units. The x-coordinate...

1. TRANSLATE the problem information

• Given information:
  • Circle centered at origin with radius 17
  • Point P lies on this circle
  • x-coordinate of P is -8
  • Need the positive y-coordinate

2. TRANSLATE the geometric relationship into algebra

• Since P lies on a circle centered at the origin with radius 17:
  • The equation is: \(\mathrm{x^2 + y^2 = r^2}\)
  • Substituting r = 17: \(\mathrm{x^2 + y^2 = 17^2 = 289}\)

3. SIMPLIFY by substituting the known x-coordinate

• Substitute x = -8:
  \(\mathrm{(-8)^2 + y^2 = 289}\)
  \(\mathrm{64 + y^2 = 289}\)
  \(\mathrm{y^2 = 289 - 64 = 225}\)

4. SIMPLIFY by taking the square root

• Take the square root: \(\mathrm{y = ±\sqrt{225} = ±15}\)

5. APPLY CONSTRAINTS to select the final answer

• The problem asks for the positive value
• Therefore: \(\mathrm{y = 15}\)

Answer: 15

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not recognize that "point P lies on a circle centered at the origin" translates to the equation \(\mathrm{x^2 + y^2 = r^2}\). Instead, they might try to use distance formula with two points or get confused about what mathematical relationship to set up.

This leads to confusion and abandoning systematic solution in favor of guessing.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly solve to get \(\mathrm{y = ±15}\) but fail to notice the problem specifically asks for the "positive value." They might submit \(\mathrm{-15}\) or become uncertain about which value to choose.

While this won't lead to selecting a wrong multiple choice answer (since only 15 would be among the choices), it reflects incomplete problem comprehension.

The Bottom Line:

This problem tests whether students can translate geometric language into algebraic equations. The key insight is recognizing that being "on a circle" means satisfying the circle equation - once that connection is made, the algebra is straightforward.