SAT Math Practice Question #787 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #787

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 is given by the equation \[ x^2 + y^2 - 14x - 6y + 33 = 0 \] Line \( L \) is tangent to the circle at the point \( (10, 7) \). Line \( M \) is parallel to line \( L \) and is also tangent to the circle. What is the \( y \)-intercept of line \( M \)?
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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

To solve this 800-level problem, we need to utilize completing the square, tangent line properties, and point symmetry! Step 1: Find the center and radius of the circle. We begin by converting the circle's equation from standard form to cen...

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

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