SAT Math: Circle (x - 3)² + (y + 2)² = 25 Tangent to Line L at (6, 2), Find Slope

Exact Question Text

In the xy-plane, the circle (x - 3)^2 + (y + 2)^2 = 25 is tangent to line L at (6, 2). What is the slope of line L?

1. Step-by-Step Formal Solution

1. Center = (3, -2), Point = (6, 2).
2. m_radius = (2 - (-2)) / (6 - 3) = 4/3.
3. Perpendicular slope m_tangent = -1 / (4/3) = -3/4.

2. Fast Desmos / Calculator Shortcut

Plot circle and point (6, 2). Calculate -(6-3)/(2 - (-2)) = -3/4.

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.