Question:Which of the following equations represents a circle in the xy-plane that does not intersect the x-axis?

1. TRANSLATE the problem requirements

• We need a circle that does NOT intersect the x-axis
• This means the circle is entirely above or entirely below the x-axis
• Each option is in standard form: \((x - h)² + (y - k)² = r²\)

2. INFER the geometric relationship

• For a circle not to intersect the x-axis, the distance from its center to the x-axis must be greater than its radius
• Distance from center \((h, k)\) to x-axis = \(|k|\) (absolute value of y-coordinate)
• Required condition: \(|k| \gt r\)

3. TRANSLATE each equation to find centers and radius

• All equations have 25 on the right side, so radius \(r = \sqrt{25} = 5\)
• Extract centers from each equation:
  • (A): Center \((6, 0)\)
  • (B): Center \((3, 4)\)
  • (C): Center \((4, 5)\)
  • (D): Center \((5, 6)\)

4. APPLY CONSTRAINTS to test each option

Check if \(|k| \gt 5\) for each center:

• Option A: \(|0| = 0\). Is \(0 \gt 5\)? No.
• Option B: \(|4| = 4\). Is \(4 \gt 5\)? No.
• Option C: \(|5| = 5\). Is \(5 \gt 5\)? No. (This means tangent to x-axis)
• Option D: \(|6| = 6\). Is \(6 \gt 5\)? Yes! ✓

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students may not connect "does not intersect x-axis" to the geometric condition about distance from center to axis. They might try to solve for intersection points algebraically by setting \(y = 0\), leading to complex calculations without clear direction. This leads to confusion and guessing.

Second Most Common Error:

Inadequate APPLY CONSTRAINTS execution: Students understand the \(|k| \gt r\) condition but make errors in checking. They might conclude that \(|5| \gt 5\) is true (confusing ≥ with >), leading them to select Choice C. Or they might forget that distance to x-axis uses absolute value, potentially making sign errors.

The Bottom Line:

This problem requires spatial reasoning to translate the intersection condition into a distance relationship. Students who try to work algebraically instead of geometrically often get overwhelmed by the calculations.