The equation x^2 + y^2 - 6x + 8y = 0 defines a circle in the xy-plane. What is the...

1. TRANSLATE the problem information

• Given information:
  • Circle equation: \(\mathrm{x^2 + y^2 - 6x + 8y = 0}\)
  • Need to find: radius of the circle

2. INFER the approach

• The equation is in general form, but we need standard form to identify the radius
• Standard form is \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\), where r is the radius
• We'll need to complete the square for both x and y terms

3. SIMPLIFY by rearranging terms

• Start with: \(\mathrm{x^2 + y^2 - 6x + 8y = 0}\)
• Rearrange: \(\mathrm{x^2 - 6x + y^2 + 8y = 0}\)

4. SIMPLIFY by completing the square for x terms

• For \(\mathrm{x^2 - 6x}\): take half of -6, which gives -3
• Square it: \(\mathrm{(-3)^2 = 9}\)
• So: \(\mathrm{x^2 - 6x + 9 = (x - 3)^2}\)

5. SIMPLIFY by completing the square for y terms

• For \(\mathrm{y^2 + 8y}\): take half of 8, which gives 4
• Square it: \(\mathrm{4^2 = 16}\)
• So: \(\mathrm{y^2 + 8y + 16 = (y + 4)^2}\)

6. SIMPLIFY by substituting and solving

• Replace in the equation: \(\mathrm{(x - 3)^2 - 9 + (y + 4)^2 - 16 = 0}\)
• Combine constants: \(\mathrm{(x - 3)^2 + (y + 4)^2 - 25 = 0}\)
• Move constant to right side: \(\mathrm{(x - 3)^2 + (y + 4)^2 = 25}\)

7. INFER the radius from standard form

• The equation \(\mathrm{(x - 3)^2 + (y + 4)^2 = 25}\) is now in standard form
• Comparing to \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\), we have \(\mathrm{r^2 = 25}\)
• Therefore: \(\mathrm{r = \sqrt{25} = 5}\)

Answer: C) 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make sign errors or forget to add/subtract the completing-the-square constants correctly.

For example, they might write \(\mathrm{(x - 3)^2 + (y + 4)^2 = -25}\) instead of +25 by incorrectly handling the -9 and -16 terms. This leads to confusion since negative values don't work for \(\mathrm{r^2}\), causing them to get stuck and guess.

Second Most Common Error:

Inadequate INFER reasoning: Students may not recognize they need to complete the square, instead trying to factor or use other approaches that don't work with this form.

Without the right strategy, they waste time on incorrect approaches and end up guessing. This leads to randomly selecting answers like Choice A (3) or Choice B (4) based on seeing familiar numbers in the original equation.

The Bottom Line:

This problem tests whether students can systematically convert between forms of circle equations. Success requires both recognizing the right algebraic technique and executing it accurately through multiple steps.