SAT Math: A Circle Has Diameter Endpoints at (-3, 2) and (5, 8), Find Its Equation

Exact Question Text

A circle in the xy-plane has a diameter with endpoints at (-3, 2) and (5, 8). What is the standard equation of the circle?

1. Step-by-Step Formal Solution

1. Center = ((-3+5)/2, (2+8)/2) = (1, 5).
2. Radius r = sqrt((5 - 1)^2 + (8 - 5)^2) = sqrt(25) = 5.
3. Equation: (x - 1)^2 + (y - 5)^2 = 25.

2. Fast Desmos / Calculator Shortcut

Plot points (-3, 2) and (5, 8) with midpoint (1, 5). Type (x - 1)^2 + (y - 5)^2 = 25 to verify it passes through both endpoints.

3. Distractor Trap Analysis

• Trap A (Sign Flip): Forgetting to reverse inequality or dropping negative sign.
• Trap B (Partial Evaluation): Solving for intermediate variable and stopping early.
• Trap C (Extreme Assumption): Over-inferring beyond textual evidence.
• Option Correct: Accurately satisfies all algebraic and logical constraints.