In the xy-plane, the circle with equation \(\mathrm{x^2 + y^2 = (a + c)^2}\) intersects the horizontal line y =...

1. TRANSLATE the problem information

• Given information:
  • Circle equation: \(\mathrm{x^2 + y^2 = (a + c)^2}\)
  • Horizontal line: \(\mathrm{y = c}\)
  • Both a and c are positive constants
• We need to find where these two curves intersect

2. TRANSLATE the intersection approach

• To find intersection points, substitute the line equation into the circle equation
• Replace y with c in \(\mathrm{x^2 + y^2 = (a + c)^2}\)

3. SIMPLIFY the resulting equation

• Substituting \(\mathrm{y = c}\): \(\mathrm{x^2 + c^2 = (a + c)^2}\)
• Expand the right side: \(\mathrm{x^2 + c^2 = a^2 + 2ac + c^2}\)
• Subtract \(\mathrm{c^2}\) from both sides: \(\mathrm{x^2 = a^2 + 2ac}\)

4. INFER the nature of solutions

• Since \(\mathrm{a \gt 0}\) and \(\mathrm{c \gt 0}\), we know:
  • \(\mathrm{a^2 \gt 0}\) (squares of positive numbers are positive)
  • \(\mathrm{2ac \gt 0}\) (product of positive numbers is positive)
  • Therefore \(\mathrm{a^2 + 2ac \gt 0}\)

5. SIMPLIFY to find x-values

• Since \(\mathrm{x^2 = a^2 + 2ac}\) and \(\mathrm{a^2 + 2ac \gt 0}\), we can take the square root
• \(\mathrm{x = \pm\sqrt{a^2 + 2ac}}\)
• This gives us exactly two distinct x-values

6. INFER the final answer

• Two x-values with the same y-value (\(\mathrm{y = c}\)) means two intersection points:
  • \(\mathrm{(\sqrt{a^2 + 2ac}, c)}\) and \(\mathrm{(-\sqrt{a^2 + 2ac}, c)}\)

Answer: C (Two)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students fail to recognize that \(\mathrm{a^2 + 2ac}\) is always positive when \(\mathrm{a \gt 0}\) and \(\mathrm{c \gt 0}\).

They might look at \(\mathrm{x^2 = a^2 + 2ac}\) and think "this could be negative" or worry about whether solutions exist, leading to unnecessary confusion about whether the square root is real. Without confidently concluding that \(\mathrm{a^2 + 2ac \gt 0}\), they may incorrectly think there are no intersection points or guess randomly.

This may lead them to select Choice A (Zero) or causes confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(a + c)^2}\).

They might incorrectly expand to \(\mathrm{a^2 + c^2}\) instead of \(\mathrm{a^2 + 2ac + c^2}\), leading to \(\mathrm{x^2 = 0}\) after substitution and cancellation. This would suggest only one intersection point at \(\mathrm{x = 0}\).

This may lead them to select Choice B (One).

The Bottom Line:

This problem requires students to systematically work through algebraic substitution while maintaining confidence in their constraint analysis. The key insight is recognizing that positive constants guarantee positive expressions under certain operations.