In the xy-plane, the set of points \(\mathrm{(x, y)}\) satisfying x^2 + y^2 - 8x + 12y = t, where...

1. TRANSLATE the problem information

• Given information:
  • Circle equation: \(\mathrm{x^2 + y^2 - 8x + 12y = t}\)
  • Point \(\mathrm{(-2, 1)}\) lies on this circle
  • Need to find the radius length

2. INFER the approach

• Key insight: The given equation isn't in standard circle form \(\mathrm{(x-h)^2 + (y-k)^2 = r^2}\)
• Strategy: Complete the square to reveal center and radius, then use the given point to find the specific value

3. SIMPLIFY by completing the square

• Start with: \(\mathrm{x^2 + y^2 - 8x + 12y = t}\)
• Group like terms: \(\mathrm{x^2 - 8x + y^2 + 12y = t}\)
• Complete the square for x:
  • Take half of −8: \(\mathrm{(-8)/2 = -4}\)
  • Square it: \(\mathrm{(-4)^2 = 16}\)
  • So \(\mathrm{x^2 - 8x + 16 = (x - 4)^2}\)
• Complete the square for y:
  • Take half of 12: \(\mathrm{12/2 = 6}\)
  • Square it: \(\mathrm{6^2 = 36}\)
  • So \(\mathrm{y^2 + 12y + 36 = (y + 6)^2}\)

4. SIMPLIFY to get standard form

• \(\mathrm{(x^2 - 8x + 16) + (y^2 + 12y + 36) = t + 16 + 36}\)
• \(\mathrm{(x - 4)^2 + (y + 6)^2 = t + 52}\)

Now we can see: center = \(\mathrm{(4, -6)}\) and \(\mathrm{r^2 = t + 52}\)

5. INFER how to find t using the given point

• Since \(\mathrm{(-2, 1)}\) lies on the circle, it must satisfy the equation
• Substitute into the original equation: \(\mathrm{(-2)^2 + (1)^2 - 8(-2) + 12(1) = t}\)

6. SIMPLIFY to calculate t

• \(\mathrm{4 + 1 + 16 + 12 = t}\)
• \(\mathrm{t = 33}\)

7. SIMPLIFY to find the radius

• \(\mathrm{r^2 = t + 52 = 33 + 52 = 85}\)
• \(\mathrm{r = \sqrt{85}}\)

Answer: \(\mathrm{\sqrt{85}}\) (Choice B)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that completing the square is necessary to identify the circle's center and radius relationship.

Students often try to work directly with the original equation \(\mathrm{x^2 + y^2 - 8x + 12y = t}\), attempting to substitute the point \(\mathrm{(-2, 1)}\) without first converting to standard form. They might calculate \(\mathrm{t = 33}\) correctly but then get stuck because they can't see how to extract the radius from the original form. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors when completing the square, particularly with the y-terms.

Students frequently write \(\mathrm{(y - 6)^2}\) instead of \(\mathrm{(y + 6)^2}\) because they forget that \(\mathrm{y^2 + 12y}\) requires adding \(\mathrm{(12/2)^2 = 36}\) to get \(\mathrm{(y + 6)^2}\). This error changes the center coordinates and leads to an incorrect radius calculation. This may lead them to select one of the other answer choices or abandon the systematic approach.

The Bottom Line:

This problem tests whether students can bridge the gap between a general circle equation and the geometric properties they need. The key breakthrough is recognizing that completing the square isn't just algebraic manipulation—it's the tool that reveals the circle's center, which then allows radius calculation.