In the xy-plane, circle A has equation x^2 + y^2 - 10x + 4y + 13 = 0. Circle B...

1. SIMPLIFY the equation to standard form

The given equation \(\mathrm{x^2 + y^2 - 10x + 4y + 13 = 0}\) is in general form. To find the center, we need standard form \(\mathrm{(x - h)^2 + (y - k)^2 = r^2}\).

• Complete the square for x terms: \(\mathrm{x^2 - 10x}\)
  • Take half of -10: \(\mathrm{(-10)/2 = -5}\)
  • Square it: \(\mathrm{(-5)^2 = 25}\)
  • Add and subtract: \(\mathrm{(x^2 - 10x + 25) - 25 = (x - 5)^2 - 25}\)

• Complete the square for y terms: \(\mathrm{y^2 + 4y}\)
  • Take half of 4: \(\mathrm{4/2 = 2}\)
  • Square it: \(\mathrm{(2)^2 = 4}\)
  • Add and subtract: \(\mathrm{(y^2 + 4y + 4) - 4 = (y + 2)^2 - 4}\)

• Rearrange: \(\mathrm{(x - 5)^2 - 25 + (y + 2)^2 - 4 + 13 = 0}\)
• Simplify: \(\mathrm{(x - 5)^2 + (y + 2)^2 = 16}\)

2. INFER the center and radius

From standard form \(\mathrm{(x - 5)^2 + (y + 2)^2 = 16}\):

• Center of Circle A: \(\mathrm{(5, -2)}\)
• Radius: \(\mathrm{\sqrt{16} = 4}\)

3. TRANSLATE the movement directions

• "3 units to the left" means x-coordinate decreases by 3
• "5 units up" means y-coordinate increases by 5

4. APPLY the translation to find new center

• New x-coordinate: \(\mathrm{5 - 3 = 2}\)
• New y-coordinate: \(\mathrm{-2 + 5 = 3}\)
• Circle B center: \(\mathrm{(2, 3)}\)
• Radius stays the same: 4

5. Write Circle B's equation

\(\mathrm{(x - 2)^2 + (y - 3)^2 = 16}\)

Answer: (A)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students struggle with completing the square, especially when dealing with both x and y terms simultaneously. They might make algebraic errors when moving terms around or forget to add the completing-the-square constants to both sides.

This leads to an incorrect center for Circle A, which propagates through the entire solution. They end up with wrong coordinates and may select Choice (D) (\(\mathrm{(x - 8)^2 + (y + 7)^2 = 16}\)) if they somehow get confused about the original center.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret the direction of translation. They might think "3 units left" means adding 3 to the x-coordinate, or "5 units up" means subtracting 5 from the y-coordinate.

With the correct center \(\mathrm{(5, -2)}\) but wrong translation direction, they might get center \(\mathrm{(8, -7)}\), leading them to select Choice (D) (\(\mathrm{(x - 8)^2 + (y + 7)^2 = 16}\)).

The Bottom Line:

This problem combines two distinct skills - algebraic manipulation (completing the square) and geometric transformation (translation). Students need to be solid on both to succeed, and weakness in either area derails the solution.