A circle in the xy-plane has its center at \((-1, 1)\). Line ell is tangent to this circle at the...

1. TRANSLATE the problem information

• Given information:
  • Circle center: \((-1, 1)\)
  • Line ℓ is tangent to circle at point \((5, -4)\)
  • Need to find another point on line ℓ

2. INFER the key geometric relationship

• Since line ℓ is tangent to the circle at \((5, -4)\), there's a crucial perpendicular relationship:
  • The radius from center \((-1, 1)\) to tangent point \((5, -4)\) is perpendicular to line ℓ
• Strategy: Find the radius slope first, then use the negative reciprocal to get the tangent line slope

3. SIMPLIFY to find the radius slope

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• Radius slope = \(\frac{-4 - 1}{5 - (-1)} = -\frac{5}{6}\)

4. INFER the tangent line slope

• Since perpendicular lines have slopes that are negative reciprocals:
• Tangent line slope = \(-\frac{1}{-\frac{5}{6}} = \frac{6}{5}\)

5. SIMPLIFY to test each answer choice

• For a point to lie on line ℓ, the slope from \((5, -4)\) to that point must equal \(\frac{6}{5}\)

Testing each choice:

• A. \((0, \frac{8}{5})\): slope = \(\frac{\frac{8}{5} + \frac{20}{5}}{-5} = -\frac{28}{25}\) ✗
• B. \((4, 7)\): slope = \(\frac{11}{-1} = -11\) ✗
• C. \((10, 2)\): slope = \(\frac{6}{5}\) ✓
• D. \((11, 1)\): slope = \(\frac{5}{6}\) ✗

Answer: C. \((10, 2)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the perpendicular relationship between the tangent line and radius. Instead, they might try to use the center and tangent point to directly find the tangent line slope, calculating \(-\frac{5}{6}\) instead of \(\frac{6}{5}\). This leads them to select Choice D. \((11, 1)\) since the slope from \((5, -4)\) to \((11, 1)\) is \(\frac{5}{6}\), which they incorrectly think matches their calculated \(-\frac{5}{6}\).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need the negative reciprocal of \(-\frac{5}{6}\), but make arithmetic errors when calculating \(\frac{6}{5}\) or when computing slopes to test answer choices. The negative numbers and fractions create multiple opportunities for sign errors or computational mistakes, leading to confusion and guessing.

The Bottom Line:

This problem requires connecting geometric relationships (perpendicular tangent/radius) with algebraic calculations (slopes and negative reciprocals). Students who memorize formulas without understanding the underlying geometric reasoning will struggle to set up the problem correctly.