SAT Math Practice Question #328 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #328

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

The equation \( x^2 + y^2 - 2ax + 4by + c = 0 \) represents a circle in the xy-plane, where \( a \), \( b \), and \( c \) are positive constants. The circle is tangent to the x-axis and has a radius of \( 8 \). If the center of the circle lies on the line \( y = -2x + 2 \), what is the value of \( a + b + c \)?
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Socratic AI Engine Step-by-Step Derivation
0ms Precomputed
Tactical Insight (Hint 1)

Identify the quadratic form: consider converting to vertex form \( y = a(x-h)^2 + k \) or using \( x = - rac{b}{2a} \) to find the line of symmetry.

Elimination Framework (Hint 2)

Desmos Tactical Shortcut: Plot the left-hand side and right-hand side as separate equations (e.g. \( y_1 = f(x) \) and \( y_2 = g(x) \)). Click the intersection points directly to read exact coordinates.

Masterclass Solution & Distractor Trap Analysis

Get ready, this is a beautiful synthesis of algebra and coordinate geometry! Let's conquer it step by step. Step 1: Find the center of the circle. First, we complete the square for the circle's equation: \( x^2 - 2ax + y^2 + 4by = -c \) Add...

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

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