Which of the following equations represents a circle in the xy-plane that intersects the y-axis at exactly one point?

1. TRANSLATE the problem requirement

• Given: Four circle equations in standard form
• Need: Circle that intersects y-axis at exactly one point
• INFER what this means: "Intersects at exactly one point" = tangent to the y-axis

2. INFER the geometric condition for tangency

• For a circle to be tangent to the y-axis, the distance from its center to the y-axis must equal its radius
• Distance from point (h, k) to y-axis = |h|
• Therefore: For tangency to y-axis, we need \(|h| = r\)

3. TRANSLATE each equation to find center and radius

For standard form \((x - h)^2 + (y - k)^2 = r^2\):

Choice A: \((x - 8)^2 + (y - 8)^2 = 16\)
• Center: (8, 8), radius = \(\sqrt{16} = 4\)

Choice B: \((x - 8)^2 + (y - 4)^2 = 16\)
• Center: (8, 4), radius = 4

Choice C: \((x - 4)^2 + (y - 9)^2 = 16\)
• Center: (4, 9), radius = 4

Choice D: \(x^2 + (y - 9)^2 = 16\)
• Center: (0, 9), radius = 4

4. APPLY CONSTRAINTS using the tangency condition

Check if \(|h| = r\) for each circle:

• Choice A: \(|8| = 8\), but \(r = 4\) → \(8 \neq 4\) ✗
• Choice B: \(|8| = 8\), but \(r = 4\) → \(8 \neq 4\) ✗
• Choice C: \(|4| = 4\), and \(r = 4\) → \(4 = 4\) ✓
• Choice D: \(|0| = 0\), but \(r = 4\) → \(0 \neq 4\) ✗

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect "intersects y-axis at exactly one point" with the concept of tangency. Instead, they might try to substitute \(x = 0\) into each equation and count intersection points algebraically, which is much more complex and time-consuming. This approach often leads to calculation errors and confusion about what the results mean geometrically.

This leads to confusion and guessing among the choices.

Second Most Common Error:

Conceptual confusion about distance: Students might incorrectly think the distance from center (h, k) to the y-axis is k (the y-coordinate) instead of |h| (absolute value of x-coordinate). This fundamental misunderstanding about coordinate geometry makes it impossible to set up the correct tangency condition.

This may lead them to incorrectly analyze the equations and select Choice D since it has the center on the y-axis, thinking this creates tangency.

The Bottom Line:

This problem tests whether students can connect geometric concepts (tangency) with algebraic representations (circle equations). The key insight is recognizing that tangency creates a specific distance relationship, not just solving for intersection points algebraically.