SAT Math Practice Question #677 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #677

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

A circle in the \( xy \)-plane has the equation \[ x^2 + y^2 - 14x + 6y + k = 0 \] where \( k \) is a constant. If the circle is tangent to the line \( 3x - 4y + 2 = 0 \), what is the value of \( k \)?
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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 rigorous Advanced Math question combining circle equations with line tangency. There are two ways to solve it: the geometric distance method and the algebraic discriminant method. We will use the geometric method as it is more ele...

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

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