Mathematics · Vectors and Conic Sections

Let vector c and vector d be vectors such that |vector c + vector d| = √29 and vector c × (2î + 3ĵ + 4k̂) = (2î + 3ĵ + 4k̂) × vector d. If λ1, λ2 (λ1 > λ2) are the possible values of (vector c + vector d) · (-7î + 2ĵ + 3k̂), then the equation K2x2 + (K2 - 5K + λ1)xy + (3K + λ22)y2 - 8x + 12y + λ2 = 0 represents a circle, for K equal to:

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