In the coordinate plane, point A has coordinates \((5, -2)\). Point B is obtained by moving point A to the...

1. TRANSLATE the coordinate movement

• Given information:
  • Point A is at \((5, -2)\)
  • Point B is obtained by moving A "to the left 7 units"
  • Moving left means subtracting from the x-coordinate
• Point B coordinates: \((5 - 7, -2) = (-2, -2)\)

2. INFER what equation form to use

• We need a circle centered at point B with radius 3
• This calls for the standard circle equation: \((x - h)^2 + (y - k)^2 = r^2\)
• Where \((h, k)\) is the center and \(r\) is the radius

3. SIMPLIFY by substituting values

• Center: \((h, k) = (-2, -2)\)
• Radius: \(r = 3\)
• Substitute into standard form:
  • \((x - (-2))^2 + (y - (-2))^2 = 3^2\)
  • \((x + 2)^2 + (y + 2)^2 = 9\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly interpret "moving left 7 units." Some add 7 instead of subtracting, thinking of moving in the positive direction, getting point B at \((12, -2)\) instead of \((-2, -2)\).

This leads them to write the equation as \((x - 12)^2 + (y + 2)^2 = 9\), causing them to select Choice B.

Second Most Common Error:

Poor SIMPLIFY execution: Students find the correct center \((-2, -2)\) but make sign errors when substituting into the standard form. They might write \((x - (-2))^2\) as \((x - 2)^2\) instead of \((x + 2)^2\), or confuse which coordinate is which.

This confusion leads to selecting incorrect equations like Choice C or Choice D, or causes them to get stuck and guess.

The Bottom Line:

This problem tests whether students can accurately translate geometric movements into coordinate changes and then carefully handle negative signs in algebraic expressions. The key insight is that moving left means subtracting from x, and substituting negative coordinates requires careful attention to sign changes.