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If the line y - 2 = 0 is the directrix of the parabola x 2 - k y + 32 = 0 , k ≠ 0 and the parabola intersects the circle x 2 + y 2 = 8 at two real distinct points, then the absolute value of k is

Correct answer

16

Step-by-step solution

x 2 - k y + 32 = 0 ⇒ x 2 = k y - 32 k Put, x = X , y - 32 k = Y The equation of directrix is Y + k 4 = 0 i.e. y - 32 k + k 4 = 0 But, y - 2 = 0 is the directrix. ⇒ 32 k - k 4 = 2 ⇒ k 2 + 8 k - 128 = 0 ⇒ k = - 16 or k = 8 For k = 8 , the parabola is x 2 = 8 y - 4 which does not intersect the circle. For k = - 16 , the parabola is x 2 = - 16 y + 2 which intersects the circle at two real distinct points. ⇒ Absolute value of k = - 16 = 16

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