Question:A circle in the xy-plane is defined by the equation \((\mathrm{x} + 10)^2 + (\mathrm{y} - 8)^2 = 100\). What...

1. TRANSLATE the equation to standard form

• Given equation: \((\mathrm{x} + 10)^2 + (\mathrm{y} - 8)^2 = 100\)
• Standard form of circle: \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• The center is at point \((\mathrm{h}, \mathrm{k})\)

2. INFER the center coordinates by comparing formats

• For the x-term: \((\mathrm{x} + 10)^2\) needs to match \((\mathrm{x} - \mathrm{h})^2\)
• Since \((\mathrm{x} + 10) = (\mathrm{x} - (-10))\), we have \(\mathrm{h} = -10\)
• For the y-term: \((\mathrm{y} - 8)^2\) already matches \((\mathrm{y} - \mathrm{k})^2\)
• So \(\mathrm{k} = 8\)

3. TRANSLATE the mathematical result to coordinates

• Center coordinates: \((\mathrm{h}, \mathrm{k}) = (-10, 8)\)

Answer: B \((-10, 8)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly handle the signs when matching to standard form.

They see \((\mathrm{x} + 10)^2\) and think \(\mathrm{h} = +10\), not recognizing that \((\mathrm{x} + 10)^2 = (\mathrm{x} - (-10))^2\). This leads them to identify the center as \((10, 8)\) instead of \((-10, 8)\).

This may lead them to select Choice D \((10, 8)\).

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students get confused about which coordinate corresponds to which variable.

They might correctly find \(\mathrm{h} = -10\) and \(\mathrm{k} = 8\), but then mix up the order and report the center as \((8, -10)\) or make other coordinate ordering errors. While this specific error doesn't match any given choice exactly, it causes confusion and may lead to guessing.

The Bottom Line:

The key challenge is carefully tracking the signs when converting between \((\mathrm{x} + \mathrm{a})^2\) and \((\mathrm{x} - \mathrm{h})^2\) formats. The equation gives you the center directly, but only if you handle the algebra notation correctly.