The equation \((\mathrm{x} - 6)^2 + (\mathrm{y} + 5)^2 = 75\) is an equation of a circle in the xy-plane....

1. TRANSLATE the equation format

• Given equation: \((\mathrm{x} - 6)^2 + (\mathrm{y} + 5)^2 = 75\)
• This matches the standard form: \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• From this match: \(\mathrm{r}^2 = 75\)

2. INFER what we need to find

• The problem asks for the diameter
• We have \(\mathrm{r}^2\), so we need to find \(\mathrm{r}\) first, then use \(\mathrm{d} = 2\mathrm{r}\)
• Next step: solve \(\mathrm{r}^2 = 75\)

3. SIMPLIFY the radical

• \(\mathrm{r} = \sqrt{75}\)
• Factor 75 to find perfect squares: \(75 = 25 \times 3\)
• \(\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}\)
• So the radius is \(5\sqrt{3}\)

4. INFER the final calculation needed

• We found the radius, but need the diameter
• Use the relationship: diameter = 2 × radius
• \(\mathrm{d} = 2(5\sqrt{3}) = 10\sqrt{3}\)

Answer: B) \(10\sqrt{3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't immediately recognize the standard form of a circle equation or confuse what \(\mathrm{r}^2\) represents in the equation.

They might think \(\mathrm{r}^2 = 75\) means \(\mathrm{r} = 75\), completely skipping the square root step. This leads them to calculate diameter as \(2(75) = 150\), causing them to select Choice D (150).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{r} = \sqrt{75}\) but fail to simplify the radical properly.

They might leave the answer as \(\mathrm{r} = \sqrt{75}\) and calculate diameter as \(2\sqrt{75}\). Since this doesn't match any answer choice exactly, they get confused and may guess or incorrectly estimate \(\sqrt{75} \approx 9\), leading to diameter ≈ 18, which might push them toward Choice C (75) as the 'closest' large number.

The Bottom Line:

This problem tests whether students can connect the abstract standard form of a circle equation to concrete geometric measurements. The key breakthrough is recognizing that the constant term gives you \(\mathrm{r}^2\), not \(\mathrm{r}\) directly.