The angle between the lines x-1 l = y+1 m = z n and x+1 m = y-3 n = z-1 l , where l m n and l , m , n are roots of the equation x^3+x^2-4 x-4=0 , is

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The angle between the lines x-1 l = y+1 m = z n and x+1 m = y-3 n = z-1 l , where l> m > n and l , m , n are roots of the equation x^3+x^2-4 x-4=0 , is

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Correct answer: B. ⁻¹ ( -4 9 )

To find the angle between the lines, begin by solving the cubic equation x^3 + x^2 - 4x - 4 = 0 . By factoring, this becomes (x^2 - 4)(x + 1) = (x - 2)(x + 2)(x + 1) with roots 2 , -2 , and -1 . Since l > m > n , assign l = 2 , m = -1 , and n = -2 . The direction vectors of the lines are then: d₁ = (2, -1, -2) and d₂ = (-1, -2, 2) . The dot product d₁ d₂ = 2 (-1) + (-1) (-2) + (-2) 2 = -4 , and both magnitudes equal 9 = 3 . The cosine of the angle is therefore -4 9 , giving = ⁻¹ ( -4 9 ) , matching option B. Final

The angle between the lines x-1 l = y+1 m = z n and x+1 m = y-3 n = z-1 l , where l> m > n and l , m , n are roots of the equation x^3+x^2-4 x-4=0 , is

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