2x^2 + 2y^2 - 20x = -8y + 30 The equation above represents a circle in the xy-plane. The center...

1. TRANSLATE the problem information

• Given equation: \(2\mathrm{x}^2 + 2\mathrm{y}^2 - 20\mathrm{x} = -8\mathrm{y} + 30\)
• Need to find: h (x-coordinate of center)

2. INFER the approach needed

• To find the center, we need the equation in standard form: \(\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• This requires completing the square for both x and y terms
• First, let's rearrange and simplify the equation

3. SIMPLIFY by rearranging and reducing

• Move all terms to one side: \(2\mathrm{x}^2 + 2\mathrm{y}^2 - 20\mathrm{x} + 8\mathrm{y} = 30\)
• Divide everything by 2 to simplify: \(\mathrm{x}^2 + \mathrm{y}^2 - 10\mathrm{x} + 4\mathrm{y} = 15\)

4. SIMPLIFY by completing the square

• Group the x and y terms: \(\mathrm{x}^2 - 10\mathrm{x}) + (\mathrm{y}^2 + 4\mathrm{y}) = 15\)

For x-terms:

• Take half the coefficient of x: \(-10 \div 2 = -5\)
• Square it: \((-5)^2 = 25\)
• So: \(\mathrm{x}^2 - 10\mathrm{x} + 25 = (\mathrm{x} - 5)^2\)

For y-terms:

• Take half the coefficient of y: \(4 \div 2 = 2\)
• Square it: \((2)^2 = 4\)
• So: \(\mathrm{y}^2 + 4\mathrm{y} + 4 = (\mathrm{y} + 2)^2\)

5. SIMPLIFY the final equation

• Add the completing terms to both sides:
• \(\mathrm{x} - 5)^2 + (\mathrm{y} + 2)^2 = 15 + 25 + 4 = 44\)

6. INFER the center coordinates

• From standard form \(\mathrm{x} - \mathrm{h})^2 + (\mathrm{y} - \mathrm{k})^2 = \mathrm{r}^2\)
• We can read: \(\mathrm{h} = 5\), \(\mathrm{k} = -2\)
• Therefore, the center is \((5, -2)\)

Answer: C. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when completing the square, particularly with the y-terms. They might think \(\mathrm{y}^2 + 4\mathrm{y}\) requires adding \((-2)^2 = 4\) instead of \((2)^2 = 4\), or they forget to add the completing terms to both sides of the equation.

This leads to an incorrect standard form and wrong center coordinates. For example, if they mess up the y-completion, they might get a center like \((5, 2)\) instead of \((5, -2)\), but this wouldn't directly match any answer choice, leading to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning about the approach: Students might try to find the center directly from the general form without converting to standard form first. They might incorrectly think they can just read off coefficients or use the general circle formula incorrectly.

This approach doesn't work and leaves them stuck, often leading to random answer selection among the given choices.

The Bottom Line:

This problem tests your ability to systematically convert between forms of circle equations. The key insight is recognizing that the center coordinates are only clearly visible in standard form, requiring careful algebraic manipulation to get there.