SAT Math Practice Question #467 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #467

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 \[ x^2 + (y - 3)^2 = 10 \] intersects the parabola with equation \[ y = x^2 + k \] at exactly three distinct points. What is the value of the constant \( k \)?
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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 magnificent geometry and algebra crossover! Let's visualize the scenario and execute the algebra. **Step 1: Setting up the system** We have a circle centered at \( (0, 3) \) with a radius of \( \sqrt{10} \), and an upward-facing p...

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

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