In the xy-plane, a circle with its center at the origin has a radius of 10. The line defined by...

1. TRANSLATE the geometric setup into equations

• Given information:
  • Circle centered at origin with radius 10
  • Line with equation \(\mathrm{y = x + 2}\)
  • Need positive x-coordinate of intersection

• What this tells us:
  • Circle equation: \(\mathrm{x^2 + y^2 = 100}\) (since \(\mathrm{radius^2 = 10^2 = 100}\))
  • We need to find where these two curves meet

2. INFER the solution strategy

• To find intersection points, we need both equations to be true simultaneously
• Strategy: Substitute the line equation into the circle equation
• This eliminates y and gives us a quadratic in x

3. SIMPLIFY by substituting and expanding

• Substitute \(\mathrm{y = x + 2}\) into \(\mathrm{x^2 + y^2 = 100}\):
  \(\mathrm{x^2 + (x + 2)^2 = 100}\)

• Expand the binomial:
  \(\mathrm{x^2 + x^2 + 4x + 4 = 100}\)
  \(\mathrm{2x^2 + 4x + 4 = 100}\)

• Rearrange to standard form:
  \(\mathrm{2x^2 + 4x - 96 = 0}\)
  \(\mathrm{x^2 + 2x - 48 = 0}\)

4. SIMPLIFY further by factoring the quadratic

• Look for factors of -48 that add to 2:
  \(\mathrm{8 \times (-6) = -48}\) and \(\mathrm{8 + (-6) = 2}\)

• Factor: \(\mathrm{(x + 8)(x - 6) = 0}\)

• Solutions: \(\mathrm{x = -8}\) or \(\mathrm{x = 6}\)

5. APPLY CONSTRAINTS to select the final answer

• The problem states \(\mathrm{p \gt 0}\)
• From our solutions \(\mathrm{x = -8}\) or \(\mathrm{x = 6}\), only \(\mathrm{x = 6}\) satisfies \(\mathrm{p \gt 0}\)

Answer: C (6)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to connect the geometric description to the algebraic setup. They may not realize that intersection points must satisfy both equations simultaneously, or they might not know the standard form for a circle centered at the origin.

This leads to confusion about how to start the problem and often results in guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(x + 2)^2}\) or when rearranging terms. A common mistake is writing \(\mathrm{(x + 2)^2 = x^2 + 4}\) (forgetting the middle term) which leads to the wrong quadratic equation.

This typically leads to incorrect x-values that don't match any answer choice, causing students to second-guess themselves and potentially select an arbitrary answer.

The Bottom Line:

This problem tests whether students can seamlessly connect coordinate geometry concepts with algebraic manipulation skills. The key insight is recognizing that intersection problems require solving a system of equations, which often means substitution followed by careful algebraic work.