A circle in the xy-plane has its center at \(\mathrm{(7, -2)}\). The line y = 3 is tangent to this...

1. TRANSLATE the problem information

• Given information:
  • Circle center: \((7, -2)\)
  • Tangent line: \(\mathrm{y = 3}\)
  • Circle equation: \((\mathrm{x - 7})^2 + (\mathrm{y + 2})^2 = \mathrm{k}\)
• We need to find the value of \(\mathrm{k}\)

2. INFER the key relationship

• Since \(\mathrm{y = 3}\) is tangent to the circle, it touches the circle at exactly one point
• This means the distance from the center to the tangent line equals the radius
• In the equation \((\mathrm{x - 7})^2 + (\mathrm{y + 2})^2 = \mathrm{k}\), the value \(\mathrm{k}\) represents \(\mathrm{r^2}\)

3. SIMPLIFY the distance calculation

• For horizontal line \(\mathrm{y = 3}\), distance from center \((7, -2)\) is:
  \(\mathrm{r = |y_{center} - y_{line}|}\)
  \(\mathrm{r = |(-2) - 3|}\)
  \(\mathrm{r = |-5|}\)
  \(\mathrm{r = 5}\)
• The radius is 5 units

4. INFER the final step

• Since \(\mathrm{k = r^2}\) in the circle equation:
  \(\mathrm{k = 5^2 = 25}\)

Answer: D) 25

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect the tangent line condition to the radius calculation. They may know the circle equation form but miss that tangent means "distance from center equals radius." Without this key insight, they get stuck trying to use other approaches or guess randomly.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need the distance but make calculation errors. Common mistakes include:

• Forgetting absolute value: calculating \(\mathrm{(-2) - 3 = -5}\) and using \(\mathrm{r = -5}\)
• Distance calculation error: using \(\mathrm{(7 - 3) = 4}\) instead of \(\mathrm{|(-2) - 3| = 5}\)
• Squaring error: getting \(\mathrm{r = 5}\) but calculating \(\mathrm{k = r = 5}\) instead of \(\mathrm{k = r^2 = 25}\)

These calculation errors may lead them to select Choice B (5) or other incorrect values.

The Bottom Line:

This problem tests whether students can bridge geometric concepts (tangent lines) with algebraic representations (circle equations). The key breakthrough is recognizing that "tangent" translates to "distance equals radius."