In the xy-plane, the endpoints of a diameter of a circle are \((-3, 7)\) and \((5, -1)\). Which of the...

1. TRANSLATE the problem information

• Given information:
  • Diameter endpoints: \((-3, 7)\) and \((5, -1)\)
  • Need to find circle equation in form \((x - h)² + (y - k)² = r²\)

2. INFER the approach

• To write a circle equation, we need the center \((h, k)\) and radius \(r\)
• Since we have diameter endpoints, the center is the midpoint of these points
• The radius is half the distance between these endpoints

3. Find the center using midpoint formula

• Center = \(((-3 + 5)/2, (7 + (-1))/2) = (1, 3)\)
• So \(h = 1\) and \(k = 3\)

4. SIMPLIFY to find the radius

• First calculate diameter length:
  \(\sqrt{(5-(-3))² + (-1-7)²}\)
  \(= \sqrt{8² + (-8)²}\)
  \(= \sqrt{128}\)
• SIMPLIFY the radical:
  \(\sqrt{128} = \sqrt{64×2}\)
  \(= 8\sqrt{2}\)
• Radius = diameter/2:
  \(= (8\sqrt{2})/2\)
  \(= 4\sqrt{2}\)
• Therefore:
  \(r² = (4\sqrt{2})²\)
  \(= 16×2\)
  \(= 32\)

5. Write the final equation

• Substitute into standard form: \((x - 1)² + (y - 3)² = 32\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the signs in standard form notation. They see \((x - 1)² + (y - 3)² = 32\) and think the center is at \((-1, -3)\) instead of \((1, 3)\). This happens because they don't understand that \((x - h)²\) means the center's x-coordinate is \(+h\).

This misconception leads them to work backwards from the wrong center coordinates, causing them to select Choice (A) [\((x + 1)² + (y + 3)² = 32\)].

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find the center but make arithmetic errors when calculating the radius. They might forget to divide the diameter by 2, using \(r² = 128\) instead of \(r² = 32\), or they make errors simplifying \(\sqrt{128}\).

This calculation error may lead them to select Choice (C) [\((x - 1)² + (y - 3)² = 16\)] if they incorrectly calculate \(r² = 16\).

The Bottom Line:

This problem tests whether students can connect the geometric concept of a diameter to the algebraic process of writing circle equations, requiring careful attention to both coordinate calculations and standard form notation.