SAT Math Practice Question #1158 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #1158

Test your problem-solving accuracy on this official-caliber item. Select an answer choice to check your reasoning instantly.

A circle in the \( xy \)-plane is defined by the equation \( x^2 + y^2 - 8x + 14y - 35 = 0 \). A line passing through the center of this circle has an equation \( y = 3x + b \). What is the value of \( b \)?
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Socratic AI Engine Step-by-Step Derivation
0ms Precomputed
Tactical Insight (Hint 1)

Complete the square to find the circle's center coordinates (h, k). Substitute (h, k) into the line equation.

Elimination Framework (Hint 2)

Desmos Shortcut: Paste the entire circle equation into Desmos line 1. The center point is clearly visible at (4, -7).

Masterclass Solution & Distractor Trap Analysis

Complete the square for the circle: (x^2 - 8x + 16) + (y^2 + 14y + 49) = 35 + 16 + 49 => (x - 4)^2 + (y + 7)^2 = 100. The center of the circle is (4, -7). If the line y = 3x + b passes through this center, substitute (4, -7): -7 = 3(4) + b ...

Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...

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