\((\mathrm{x} + 4)^2 + (\mathrm{y} - 19)^2 = 121\). The graph of the given equation is a circle in the...

1. TRANSLATE the equation into circle components

• Given: \(\mathrm{(x + 4)^2 + (y - 19)^2 = 121}\)
• This matches the standard circle form: \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\)
• Comparing forms:
  • \(\mathrm{(x + 4)^2 = (x - (-4))^2}\), so center x-coordinate \(\mathrm{h = -4}\)
  • \(\mathrm{(y - 19)^2}\) stays as is, so center y-coordinate \(\mathrm{k = 19}\)
  • \(\mathrm{r^2 = 121}\), so radius \(\mathrm{r = 11}\)

2. INFER what the question is really asking

• The point (a, b) lies 'on the circle' - this means a could be any x-coordinate of a point on the circle
• Key insight: We need the range of possible x-coordinates, not just any specific point
• Since the center is at x = -4 with radius 11, the circle extends 11 units left and right from center

3. SIMPLIFY to find the x-coordinate boundaries

• Leftmost point: \(\mathrm{x = center - radius = -4 - 11 = -15}\)
• Rightmost point: \(\mathrm{x = center + radius = -4 + 11 = 7}\)
• Therefore: \(\mathrm{-15 \leq a \leq 7}\)

4. APPLY CONSTRAINTS to eliminate impossible answers

• Check each choice against our range \(\mathrm{-15 \leq a \leq 7}\):
  • Choice A: \(\mathrm{-16 \lt -15}\) ❌ (too far left)
  • Choice B: -14 fits perfectly ✓ (\(\mathrm{-15 \leq -14 \leq 7}\))
  • Choice C: \(\mathrm{11 \gt 7}\) ❌ (too far right)
  • Choice D: \(\mathrm{19 \gt 7}\) ❌ (way too far right)

Answer: B. -14

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may correctly identify the center and radius but fail to recognize that they need to find the range of x-coordinates rather than solve for specific points.

They might try to solve \(\mathrm{(a + 4)^2 + (b - 19)^2 = 121}\) for specific values, getting overwhelmed by having two unknowns. Without the key insight about coordinate ranges, they resort to guessing among the choices.

This leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE execution: Students might misidentify the center coordinates, particularly getting confused by the sign in \(\mathrm{(x + 4)^2}\).

They might think the center is at \(\mathrm{(4, 19)}\) instead of \(\mathrm{(-4, 19)}\), leading to the wrong x-coordinate range of \(\mathrm{-7 \leq a \leq 18}\). With this incorrect range, choice C (11) would seem reasonable.

This may lead them to select Choice C (11).

The Bottom Line:

This problem tests whether students understand that 'lies on the circle' means finding the boundaries of where points can exist, not solving for specific coordinates. The key breakthrough is realizing that x-coordinates are constrained by the center ± radius relationship.