x^2 + 14x + y^2 = 6y + 109. In the xy-plane, the graph of the given equation is a...

1. INFER the solution approach

• Given: \(\mathrm{x^2 + 14x + y^2 = 6y + 109}\) represents a circle
• Goal: Find the radius length
• Key insight: We need to convert this to standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\) where r is the radius

2. SIMPLIFY by rearranging terms

• Move all variables to left side: \(\mathrm{x^2 + 14x + y^2 - 6y = 109}\)
• Group x terms together and y terms together

3. SIMPLIFY through completing the square

• For x terms \(\mathrm{(x^2 + 14x)}\): Add and subtract \(\mathrm{(14/2)^2 = 49}\)
  • \(\mathrm{x^2 + 14x + 49 = (x + 7)^2}\)
• For y terms \(\mathrm{(y^2 - 6y)}\): Add and subtract \(\mathrm{(-6/2)^2 = 9}\)
  • \(\mathrm{y^2 - 6y + 9 = (y - 3)^2}\)

4. SIMPLIFY the final equation

• Add the completing-the-square constants to both sides:

\(\mathrm{(x + 7)^2 + (y - 3)^2 = 109 + 49 + 9 = 167}\)

5. INFER the radius from standard form

• We now have \(\mathrm{(x + 7)^2 + (y - 3)^2 = 167}\)
• Comparing to \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\), we see \(\mathrm{r^2 = 167}\)
• Therefore: \(\mathrm{r = \sqrt{167}}\)

Answer: C. \(\mathrm{\sqrt{167}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make calculation errors when completing the square, particularly with the \(\mathrm{(b/2)^2}\) calculations or when adding constants to both sides.

For example, they might incorrectly calculate \(\mathrm{(14/2)^2}\) as something other than 49, or \(\mathrm{(-6/2)^2}\) as something other than 9. Or they might add the wrong total to the right side (getting something other than 167). These calculation errors lead to different \(\mathrm{r^2}\) values.

This may lead them to select Choice A (\(\mathrm{\sqrt{109}}\)) if they forget to add the completing-the-square constants, or Choice B (\(\mathrm{\sqrt{149}}\)) or Choice D (\(\mathrm{\sqrt{341}}\)) with other calculation mistakes.

Second Most Common Error:

Missing INFER insight: Students don't recognize that they need to complete the square to convert to standard form.

They might try to work directly with the given equation or attempt other approaches that don't lead to the standard circle form. Without completing the square, they can't identify \(\mathrm{r^2 = 167}\).

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem requires both strategic thinking (recognizing the need to complete the square) and careful algebraic execution (getting all the arithmetic right). Students who understand the strategy but make calculation errors will still get the wrong answer.