A circle in the xy-plane has its center at \((-4, 5)\) and the point \((-8, 8)\) lies on the circle....

1. TRANSLATE the problem information

• Given information:
  • Center: \((-4, 5)\)
  • Point on circle: \((-8, 8)\)
• What this tells us: We need to find the equation using the standard circle form

2. INFER the approach

• Use the standard form: \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• Substitute the center coordinates first
• Then use the point on the circle to find \(\mathrm{r}^2\)

3. TRANSLATE center coordinates into equation

• With center \((-4, 5)\): \(\mathrm{h} = -4\), \(\mathrm{k} = 5\)
• Substituting: \((\mathrm{x} - (-4))^2 + (\mathrm{y} - 5)^2 = \mathrm{r}^2\)
• SIMPLIFY: \((\mathrm{x} + 4)^2 + (\mathrm{y} - 5)^2 = \mathrm{r}^2\)

4. INFER how to find r²

• Since \((-8, 8)\) lies on the circle, these coordinates must satisfy our equation
• Substitute \(\mathrm{x} = -8\) and \(\mathrm{y} = 8\) into \((\mathrm{x} + 4)^2 + (\mathrm{y} - 5)^2 = \mathrm{r}^2\)

5. SIMPLIFY to find r²

• \((-8 + 4)^2 + (8 - 5)^2 = \mathrm{r}^2\)
• \((-4)^2 + (3)^2 = \mathrm{r}^2\)
• \(16 + 9 = \mathrm{r}^2\)
• \(25 = \mathrm{r}^2\)

6. Write the final equation

• \((\mathrm{x} + 4)^2 + (\mathrm{y} - 5)^2 = 25\)

Answer: D. \((\mathrm{x} + 4)^2 + (\mathrm{y} - 5)^2 = 25\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often get confused by the negative center coordinate and incorrectly write \((\mathrm{x} - 4)^2\) instead of \((\mathrm{x} + 4)^2\).

When they see center \((-4, 5)\), they might think: "The x-coordinate is -4, so I write \((\mathrm{x} - 4)^2\)"

But the standard form is \((\mathrm{x} - \mathrm{h})^2\), and when \(\mathrm{h} = -4\), this becomes \((\mathrm{x} - (-4))^2 = (\mathrm{x} + 4)^2\). Missing this sign relationship leads them to an equation with the wrong center, potentially selecting a choice that represents a circle centered at \((4, -5)\) instead of \((-4, 5)\).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up the equation but make arithmetic errors when calculating \(\mathrm{r}^2\).

They might compute \((-4)^2 + (3)^2\) incorrectly, perhaps getting \((-4)^2 = -16\) instead of \(+16\), or simply adding \(16 + 9\) incorrectly. This leads them to get \(\mathrm{r}^2 = 5\) instead of \(\mathrm{r}^2 = 25\), causing them to select choices with the wrong radius value.

The Bottom Line:

This problem tests whether students truly understand the relationship between center coordinates and the standard form equation, particularly handling negative coordinates correctly.