SAT Math Practice Question #827 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #827

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 with equation \( x^2 + y^2 - 10x + 6y + c = 0 \) is exactly tangent to the line \( y = 2x - 1 \). What is the value of the constant \( c \)?
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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

This is a spectacular geometry/algebra hybrid question. There are two elite ways to solve this: algebraically via the discriminant, or geometrically via the distance formula. We will use the algebra method, as it is foolproof for standard S...

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

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