SAT Math: x² + y² - 8x + 10y - 59 = 0 Tangent Line at Point P(4, 5)

Exact Question Text

The equation of a circle is x^2 + y^2 - 8x + 10y - 59 = 0. A line tangent at P(4, 5) intersects the y-axis at (0, b). What is b?

1. Step-by-Step Formal Solution

1. (x - 4)^2 + (y + 5)^2 = 100.
2. Vertical radius from (4, -5) to (4, 5).
3. Horizontal tangent y = 5 => b = 5.

2. Fast Desmos / Calculator Shortcut

Type x^2 + y^2 - 8x + 10y - 59 = 0 and plot (4, 5). Top tangent is y = 5.

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.