A circle in the xy-plane has center at \((3, -1)\). Line t is tangent to the circle at the point...

1. TRANSLATE the problem information

• Given information:
  • Circle center: \((3, -1)\)
  • Point where line is tangent: \((-5, 3)\)
  • Need to find: x-intercept of the tangent line

2. INFER the geometric relationship

• Key insight: A tangent line to a circle is always perpendicular to the radius at the point of tangency
• This means: if we find the slope of the radius, we can find the slope of the tangent line using perpendicular slopes

3. SIMPLIFY to find the radius slope

• Slope of radius from center \((3, -1)\) to point \((-5, 3)\):
  \(\mathrm{m_{radius}} = \frac{3 - (-1)}{-5 - 3}\)
  \(= \frac{4}{-8}\)
  \(= -\frac{1}{2}\)

4. INFER the tangent line slope

• Since perpendicular lines have negative reciprocal slopes:
  \(\mathrm{m_{tangent}} = -\frac{1}{-1/2} = 2\)

5. SIMPLIFY to find the tangent line equation

• Using point-slope form with point \((-5, 3)\) and slope \(2\):
  \(\mathrm{y - 3 = 2(x + 5)}\)
  \(\mathrm{y - 3 = 2x + 10}\)
  \(\mathrm{y = 2x + 13}\)

6. SIMPLIFY to find the x-intercept

• Set \(\mathrm{y = 0}\) and solve for \(\mathrm{x}\):
  \(\mathrm{0 = 2x + 13}\)
  \(\mathrm{-13 = 2x}\)
  \(\mathrm{x = -\frac{13}{2}}\)

Answer: A) \(\mathrm{-\frac{13}{2}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that the tangent line is perpendicular to the radius at the point of tangency. Students might try to find the line equation directly through two points or use other approaches that don't work with only one point given. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors when calculating the slope of the radius (especially with the negative coordinates) or when finding the negative reciprocal. For example, calculating the radius slope as \(\frac{1}{2}\) instead of \(-\frac{1}{2}\) would give a tangent slope of \(-2\), leading to the equation \(\mathrm{y = -2x - 7}\), and an x-intercept of \(-\frac{7}{2}\). This may lead them to select Choice B \((-\frac{7}{2})\).

The Bottom Line:

This problem tests whether students understand the fundamental geometric relationship between tangent lines and radii, combined with coordinate geometry skills. The key breakthrough moment is realizing that perpendicular slopes are the pathway to the solution.