SAT Math Practice Question #776 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #776

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

In the \( xy \)-plane, the system of equations is given by: \[ x^2 + y^2 - 8x + 6y + c = 0 \] \[ 3x - 4y = 12 \] where \( c \) is a constant. For what value of \( c \) does the system have exactly one real solution?
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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 an elite 800-level algebra and geometry problem. We are looking for the intersection of a circle and a line. If the system has 'exactly one real solution,' it means the line is perfectly tangent to the circle. There are two ways to ...

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

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