Question:In the xy-plane, consider the vertical line x = -3. Which of the following equations defines a circle that intersects...

1. TRANSLATE the problem information

• Given information:
  • Vertical line: \(\mathrm{x = -3}\)
  • Five circle equations in various forms
  • Need to find which circle intersects the line at exactly one point (tangent)
• What "tangent" means: The circle touches the line at exactly one point, neither missing it nor crossing through it.

2. INFER the mathematical condition for tangency

• For any circle to be tangent to a line, the distance from the circle's center to the line must equal the circle's radius
• Since we have a vertical line \(\mathrm{x = -3}\), the distance from any point \(\mathrm{(h, k)}\) to this line is \(\mathrm{|h - (-3)| = |h + 3|}\)
• Tangency condition: \(\mathrm{|h + 3| = r}\)

3. TRANSLATE each circle equation to identify center and radius

For each option, convert to standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\):

(A) \(\mathrm{(x + 3)^2 + (y - 7)^2 = 1}\) → Center: \(\mathrm{(-3, 7)}\), Radius: \(\mathrm{1}\)

(B) \(\mathrm{(x + 1)^2 + (y - 5)^2 = 4}\) → Center: \(\mathrm{(-1, 5)}\), Radius: \(\mathrm{2}\)

(C) \(\mathrm{(x - 1)^2 + (y + 2)^2 = 9}\) → Center: \(\mathrm{(1, -2)}\), Radius: \(\mathrm{3}\)

(D) \(\mathrm{(x + 5)^2 + (y + 1)^2 = 9}\) → Center: \(\mathrm{(-5, -1)}\), Radius: \(\mathrm{3}\)

4. SIMPLIFY by checking the tangency condition for each option

Calculate \(\mathrm{|h + 3|}\) for each center and compare to radius:

(A) \(\mathrm{|(-3) + 3| = 0}\), but radius \(\mathrm{= 1}\) → \(\mathrm{0 ≠ 1}\) ✗

(B) \(\mathrm{|(-1) + 3| = 2}\), and radius \(\mathrm{= 2}\) → \(\mathrm{2 = 2}\) ✓

(C) \(\mathrm{|1 + 3| = 4}\), but radius \(\mathrm{= 3}\) → \(\mathrm{4 ≠ 3}\) ✗

(D) \(\mathrm{|(-5) + 3| = 2}\), but radius \(\mathrm{= 3}\) → \(\mathrm{2 ≠ 3}\) ✗

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students know what tangency means intuitively but fail to translate it into the mathematical condition that distance equals radius.

Instead, they might try to substitute \(\mathrm{x = -3}\) into each circle equation and look for cases where the discriminant equals zero, or they might try to solve algebraically without recognizing the geometric relationship. This leads to confusion and complex calculations that don't lead anywhere productive, causing them to abandon systematic solution and guess.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misidentify the center coordinates when converting from the given form to standard form.

For example, they might think \(\mathrm{(x + 3)^2}\) means \(\mathrm{h = 3}\) instead of \(\mathrm{h = -3}\), or confuse the signs when reading the equations. This leads to calculating distances incorrectly. For instance, if they think option (A) has center \(\mathrm{(3, 7)}\), they'd calculate \(\mathrm{|3 + 3| = 6 ≠ 1}\), still concluding it's not tangent, but their reasoning process is flawed. While this particular error might not change their final answer for option (A), it could lead them to incorrectly identify option (B) as having center \(\mathrm{(1, 5)}\) and calculate \(\mathrm{|1 + 3| = 4 ≠ 2}\), causing them to eliminate the correct answer.

The Bottom Line:

This problem requires connecting geometric intuition (what tangency looks like) with algebraic manipulation (distance formulas and equation forms). Students who can't make this connection often resort to trial-and-error approaches that waste time and lead to errors.