Question:In the xy-plane, the equation 3x^2 + 3y^2 - 12x + 18y = 15 represents a circle. What is the...

1. TRANSLATE the problem information

• Given: Circle equation \(3x^2 + 3y^2 - 12x + 18y = 15\)
• Find: Distance from center to origin \((0, 0)\)
• Available answers: \(\sqrt{5}\), \(2\sqrt{2}\), \(\sqrt{10}\), \(\sqrt{13}\), 4

2. INFER the solution strategy

• To find distance from center to origin, we first need the center coordinates
• This means converting the equation to standard form: \((x - h)^2 + (y - k)^2 = r^2\)
• Then we can use the distance formula

3. SIMPLIFY by dividing by the leading coefficient

• Divide the entire equation by 3: \(x^2 + y^2 - 4x + 6y = 5\)
• This makes completing the square easier

4. SIMPLIFY using completing the square

• For x terms: \(x^2 - 4x\)
  • Take half of coefficient: \((-4)/2 = -2\)
  • Square it: \((-2)^2 = 4\)
  • So \(x^2 - 4x = (x - 2)^2 - 4\)
• For y terms: \(y^2 + 6y\)
  • Take half of coefficient: \((6)/2 = 3\)
  • Square it: \((3)^2 = 9\)
  • So \(y^2 + 6y = (y + 3)^2 - 9\)

5. SIMPLIFY to standard form

• Substitute: \((x - 2)^2 - 4 + (y + 3)^2 - 9 = 5\)
• Combine constants: \((x - 2)^2 + (y + 3)^2 = 5 + 4 + 9 = 18\)
• Center coordinates: \((2, -3)\)

6. SIMPLIFY using distance formula

• Distance = \(\sqrt{(2 - 0)^2 + (-3 - 0)^2}\)
• \(= \sqrt{4 + 9}\)
• \(= \sqrt{13}\)

Answer: D. \(\sqrt{13}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when completing the square, particularly with the y terms. They might write \((y - 3)^2\) instead of \((y + 3)^2\), leading to center coordinates of \((2, 3)\) instead of \((2, -3)\).

With center \((2, 3)\), the distance becomes \(\sqrt{4 + 9} = \sqrt{13}\), which happens to still give the right answer by coincidence. However, more commonly they calculate \(\sqrt{4 + 9}\) incorrectly or misapply the distance formula, leading to confusion and guessing.

Second Most Common Error:

Insufficient SIMPLIFY execution: Students forget to divide by 3 at the beginning, making the completing the square process much more complex with fractions. This leads to calculation errors and they may select Choice (E) 4 by approximating their messy result.

The Bottom Line:

The algebraic manipulation of completing the square with proper sign tracking is the critical hurdle. Students who can systematically complete the square will succeed, while those who rush through the algebra or make sign errors will struggle.