SAT Math Practice Question #497 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #497

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

In the \( xy \)-plane, Circle A has the equation \( x^2 - 10x + y^2 + 8y = -5 \). Circle B is formed by translating Circle A to the left by \( 4 \) units and up by \( a \) units, where \( a > 0 \). If the shortest distance between the two circles is \( 2 \), what is the value of \( a \)?
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Socratic AI Engine Step-by-Step Derivation
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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

This Geometry problem requires synthesizing circle equations, translations, and geometric distance formulas. Step 1: Find the center and radius of Circle A by completing the square. Equation: \( x^2 - 10x + y^2 + 8y = -5 \) Complete the squ...

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

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