A circle in the xy-plane has its center at \((16, 17)\) and has a radius of 7k. Which equation represents...

1. TRANSLATE the problem information

• Given information:
  • Center: \((16, 17)\)
  • Radius: \(7k\)

• What this tells us: We need to substitute these values into the standard circle equation

2. INFER the approach needed

• We need the standard form of a circle equation: \((x - h)^2 + (y - k)^2 = r^2\)
• The center \((h, k) = (16, 17)\) and radius \(r = 7k\)
• Key insight: The radius gets squared on the right side of the equation

3. TRANSLATE the values into the equation

• Substitute \(h = 16\), \(k = 17\) (noting that this k is just the y-coordinate)
• Substitute \(r = 7k\) for the radius
• This gives us: \((x - 16)^2 + (y - 17)^2 = (7k)^2\)

4. SIMPLIFY the right side

• \((7k)^2 = 7^2 \times k^2 = 49k^2\)
• Final equation: \((x - 16)^2 + (y - 17)^2 = 49k^2\)

Answer: B. \((x - 16)^2 + (y - 17)^2 = 49k^2\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students forget that the radius must be squared in the circle equation.

They might substitute \(r = 7k\) directly without squaring it, thinking the equation should be \((x - 16)^2 + (y - 17)^2 = 7k\). This leads them to select Choice C \((7k)\).

Second Most Common Error:

Poor SIMPLIFY execution: Students square only part of the radius term incorrectly.

They might compute \((7k)^2\) as \(7k^2\) instead of \(49k^2\), forgetting that \((ab)^2 = a^2b^2\). This causes them to select Choice D \((7k^2)\).

The Bottom Line:

This problem tests whether students truly understand that the standard circle equation requires \(r^2\) on the right side, not just \(r\). The algebraic manipulation of squaring \(7k\) properly is also a key skill that separates correct from incorrect solutions.