Question:x^2 - 6x + y^2 - 10y = -9The equation above defines a circle in the xy-plane. What is the...

1. INFER the approach needed

• Given information: \(\mathrm{x^2 - 6x + y^2 - 10y = -9}\)
• Strategy: Convert to standard circle form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\) to find radius
• This requires completing the square for both x and y terms

2. SIMPLIFY by grouping terms

• Group x-terms and y-terms: \(\mathrm{(x^2 - 6x) + (y^2 - 10y) = -9}\)
• This prepares us to complete the square for each variable separately

3. SIMPLIFY by completing the square for x

• For \(\mathrm{x^2 - 6x}\): Take half of coefficient → \(\mathrm{(-6)/2 = -3}\)
• Square this value: \(\mathrm{(-3)^2 = 9}\)
• Need to add 9 to make \(\mathrm{x^2 - 6x + 9 = (x - 3)^2}\)

4. SIMPLIFY by completing the square for y

• For \(\mathrm{y^2 - 10y}\): Take half of coefficient → \(\mathrm{(-10)/2 = -5}\)
• Square this value: \(\mathrm{(-5)^2 = 25}\)
• Need to add 25 to make \(\mathrm{y^2 - 10y + 25 = (y - 5)^2}\)

5. SIMPLIFY by adding constants to both sides

• Add 9 and 25 to both sides to balance the equation:
• \(\mathrm{(x^2 - 6x + 9) + (y^2 - 10y + 25) = -9 + 9 + 25}\)
• \(\mathrm{(x - 3)^2 + (y - 5)^2 = 25}\)

6. INFER the radius from standard form

• Comparing \(\mathrm{(x - 3)^2 + (y - 5)^2 = 25}\) to \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\)
• We see that \(\mathrm{r^2 = 25}\)
• Therefore \(\mathrm{r = \sqrt{25} = 5}\)

Answer: B. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic mistakes when completing the square, particularly in calculating what constants to add.

For example, they might take half of -6 incorrectly (getting -6 instead of -3) or forget to square the result. When completing the square for \(\mathrm{y^2 - 10y}\), they might calculate \(\mathrm{(-10)/2 = -5}\) correctly but then use -5 as the constant instead of \(\mathrm{(-5)^2 = 25}\). These errors lead to an incorrect final equation like \(\mathrm{(x - 3)^2 + (y - 5)^2 = 9}\), giving radius = 3.

This may lead them to select Choice A (3).

Second Most Common Error:

Poor INFER reasoning: Students don't recognize that completing the square is needed or don't understand the connection between the standard form and finding the radius.

Some students might try to work directly with the given equation or attempt to factor it like a quadratic. Others might complete the square correctly but then confuse \(\mathrm{r^2}\) with r, thinking the radius is 25 instead of 5.

This may lead them to select Choice D (25).

The Bottom Line:

This problem tests both algebraic manipulation skills and conceptual understanding of circle equations. Success requires systematic completion of the square for two variables while maintaining arithmetic accuracy throughout multiple steps.