Which of the following equations represents a circle in the xy-plane that intersects the x-axis at exactly one point?\((\mathrm{x} -...

1. TRANSLATE the key phrase

• "Intersects the x-axis at exactly one point" means the circle touches the x-axis at one point only
• This describes a circle that is tangent to the x-axis

2. INFER the geometric condition for tangency

• For a circle to be tangent to the x-axis, the distance from its center to the x-axis must equal its radius
• If the distance is less than the radius → circle crosses the x-axis at two points
• If the distance is greater than the radius → circle doesn't touch the x-axis at all
• If the distance equals the radius → circle touches at exactly one point (tangent)

3. TRANSLATE each equation to find center and radius

For circle \((\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\): center is \((\mathrm{h}, \mathrm{k})\) and radius is \(\mathrm{r}\)

• (A) Center: \((2, 3)\), radius: \(\sqrt{9} = 3\)
• (B) Center: \((1, 2)\), radius: \(\sqrt{9} = 3\)
• (C) Center: \((4, 5)\), radius: \(\sqrt{16} = 4\)
• (D) Center: \((3, 1)\), radius: \(\sqrt{4} = 2\)

4. SIMPLIFY by applying the tangency condition

Distance from center \((\mathrm{h}, \mathrm{k})\) to x-axis = \(|\mathrm{k}|\)

Check if \(|\mathrm{k}| = \mathrm{r}\) for each circle:

• (A) \(|3| = 3\) ✓ (tangent to x-axis)
• (B) \(|2| = 2 \neq 3\) (intersects at two points)
• (C) \(|5| = 5 \neq 4\) (doesn't intersect x-axis)
• (D) \(|1| = 1 \neq 2\) (intersects at two points)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "intersects at exactly one point" with the concept of tangency. They might try to solve algebraically by setting \(\mathrm{y} = 0\) in each equation and checking how many solutions exist, leading to unnecessary complex calculations and potential arithmetic errors. This leads to confusion and guessing rather than systematic elimination.

Second Most Common Error:

Missing conceptual knowledge: Students confuse the distance formula. They might calculate the distance from center to x-axis as just the y-coordinate (without absolute value) or use the full distance formula \(\sqrt{\mathrm{h}^2 + \mathrm{k}^2}\). This causes incorrect distance calculations, making them select Choice B or Choice D where their miscalculated distances happen to match the radius.

The Bottom Line:

This problem tests geometric insight more than algebraic manipulation. The key breakthrough is recognizing that "exactly one intersection point" is the definition of tangency, which has a simple geometric condition: distance equals radius.