In the xy-plane, a circle with equation \(\mathrm{x^2 + (y - k)^2 = 9}\) is tangent to the horizontal line...

1. TRANSLATE the circle equation

• Given: \(\mathrm{x² + (y - k)² = 9}\)
• This is in standard form \(\mathrm{(x - h)² + (y - k)² = r²}\)
• Center: \(\mathrm{(0, k)}\)
• Radius: \(\mathrm{√9 = 3}\)

2. INFER what tangency means

• When a circle is tangent to a line, it touches the line at exactly one point
• This happens when the distance from the circle's center to the line equals the radius
• Key insight: Distance from center to line = radius

3. TRANSLATE the distance requirement

• We need distance from point \(\mathrm{(0, k)}\) to horizontal line \(\mathrm{y = 1}\)
• For horizontal lines \(\mathrm{y = c}\), distance from \(\mathrm{(x₀, y₀)}\) is \(\mathrm{|y₀ - c|}\)
• Distance = \(\mathrm{|k - 1|}\)

4. SIMPLIFY by setting up and solving the equation

• Tangency condition: \(\mathrm{|k - 1| = 3}\)
• Absolute value equation gives us: \(\mathrm{k - 1 = 3}\) or \(\mathrm{k - 1 = -3}\)
• Solving: \(\mathrm{k = 4}\) or \(\mathrm{k = -2}\)

5. APPLY CONSTRAINTS to select final answer

• Problem states k is positive
• Therefore \(\mathrm{k = 4}\)

Answer: C) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the tangency condition

Students may know how to find the center and radius from the circle equation, but fail to connect tangency with "distance equals radius." Without this key insight, they can't set up the fundamental equation \(\mathrm{|k - 1| = 3}\). This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE reasoning: Confusing the distance setup

Some students might set up the distance incorrectly, perhaps using \(\mathrm{|k + 1| = 3}\) instead of \(\mathrm{|k - 1| = 3}\). This gives \(\mathrm{k + 1 = 3}\), so \(\mathrm{k = 2}\), leading them to select Choice B (2).

The Bottom Line:

This problem tests whether students can connect the geometric concept of tangency to its algebraic representation. The key breakthrough is recognizing that tangency translates to "distance equals radius" - without this connection, students cannot begin the solution process.