A circle in the xy-plane has its center at \((-4, 7)\). The point \((1, -5)\) lies on the circle. If...

1. TRANSLATE the problem information

• Given information:
  • Circle center: \((-4, 7)\)
  • Point on circle: \((1, -5)\)
  • Equation format: \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{c}\)
• What this tells us: We need to find the value of parameter c

2. INFER the relationship between c and circle properties

• The standard circle equation is \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• Comparing with our format \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{c}\)
• Therefore: \(\mathrm{c} = \mathrm{r}^2\) (c equals the radius squared)

3. INFER the solution strategy

• Since we know the center and one point on the circle, we can find \(\mathrm{r}^2\)
• The radius is the distance from center to any point on the circle
• We can use the distance formula squared to find c directly

4. SIMPLIFY using the distance formula squared

• Center: \((-4, 7)\), Point on circle: \((1, -5)\)
• \(\mathrm{c} = \mathrm{r}^2 = (1 - (-4))^2 + (-5 - 7)^2\)
• \(\mathrm{c} = (1 + 4)^2 + (-12)^2\)
• \(\mathrm{c} = (5)^2 + (-12)^2\)
• \(\mathrm{c} = 25 + 144\)
• \(\mathrm{c} = 169\)

Answer: D) 169

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't recognize that c represents \(\mathrm{r}^2\) in the given equation format.

They might think c represents the radius itself, not the radius squared, leading them to calculate \(\mathrm{r} = \sqrt{169} = 13\) and select Choice A (13).

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when calculating \((1 - (-4))^2 + (-5 - 7)^2\).

Common mistakes include:

• Calculating \(1 - (-4)\) as -3 instead of 5
• Getting \(-5 - 7\) as -2 instead of -12

This leads to incorrect intermediate calculations and potentially selecting Choice B (29) or getting confused and guessing.

The Bottom Line:

This problem tests whether students truly understand what each component in a circle equation represents, not just whether they can memorize the standard form. The key insight is recognizing that \(\mathrm{c} = \mathrm{r}^2\), then systematically applying the distance formula.