A circle in the xy-plane has equation \((\mathrm{x} + 3)^2 + (\mathrm{y} - 1)^2 = 25\). Which of the following...

1. TRANSLATE the circle equation

• Given: \((\mathrm{x} + 3)^2 + (\mathrm{y} - 1)^2 = 25\)
• Standard form is \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• This means: \(\mathrm{h} = -3\), \(\mathrm{k} = 1\), \(\mathrm{r}^2 = 25\)
• Center: \((-3, 1)\) and radius: \(5\)

2. INFER the solution strategy

• For a point to be in the interior (inside) of a circle, its distance to the center must be LESS than the radius
• We need to calculate the distance from each given point to center \((-3, 1)\)
• If \(\mathrm{distance} < 5\), the point is interior; if \(\mathrm{distance} \geq 5\), it's not interior

3. SIMPLIFY distance calculations using the distance formula

• Formula: \(\mathrm{d} = \sqrt{(\mathrm{x_2} - \mathrm{x_1})^2 + (\mathrm{y_2} - \mathrm{y_1})^2}\)

For each option:

Choice A: \((-7, 3)\)

• \(\mathrm{d} = \sqrt{(-7 - (-3))^2 + (3 - 1)^2}\)
• \(= \sqrt{(-4)^2 + 2^2}\)
• \(= \sqrt{16 + 4}\)
• \(= \sqrt{20} \approx 4.47\)
• Since \(4.47 < 5\), this point IS in the interior

Choice B: \((-3, 1)\)

• This is exactly the center, so \(\mathrm{distance} = 0\)
• Since \(0 < 5\), this point IS in the interior

Choice C: \((0, 0)\)

• \(\mathrm{d} = \sqrt{(0 - (-3))^2 + (0 - 1)^2}\)
• \(= \sqrt{3^2 + (-1)^2}\)
• \(= \sqrt{9 + 1}\)
• \(= \sqrt{10} \approx 3.16\)
• Since \(3.16 < 5\), this point IS in the interior

Choice D: \((3, 2)\)

• \(\mathrm{d} = \sqrt{(3 - (-3))^2 + (2 - 1)^2}\)
• \(= \sqrt{6^2 + 1^2}\)
• \(= \sqrt{36 + 1}\)
• \(= \sqrt{37} \approx 6.08\) (use calculator)
• Since \(6.08 > 5\), this point is NOT in the interior

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misidentify the center of the circle from the standard form equation. They might think the center is \((3, -1)\) instead of \((-3, 1)\) because they forget that \((\mathrm{x} + 3)^2\) means \(\mathrm{x} - (-3)\), so \(\mathrm{h} = -3\), not \(+3\).

With wrong center \((3, -1)\), they would calculate different distances and might select a different answer, leading to confusion about which point is actually outside the circle.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when calculating \(\sqrt{37}\), either approximating incorrectly or comparing wrong values to the radius. Since \(\sqrt{37} \approx 6.08\) is close to \(6\), some students might mistakenly think it's less than \(5\).

This calculation error could lead them to incorrectly conclude that point \((3, 2)\) IS in the interior, causing them to select a different choice or become confused and guess.

The Bottom Line:

This problem tests whether students can correctly extract circle parameters from standard form and accurately apply the distance formula. The key insight is that "interior" means strictly less than the radius, not less than or equal to.