SAT Math Practice Question #387 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #387

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

In the xy-plane, the circle given by the equation \( x^2 + y^2 - 12x + 8y + c = 0 \) is tangent to the line \( y = x \). What is the value of the constant \( c \)?
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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 is an exquisite 800-level geometry and algebra crossover. The key word here is 'tangent.' When a line is tangent to a curve (like a circle), it intersects that curve at exactly one point. In an algebraic system, this means if we substi...

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

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