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Consider a plane P: 2x + y - 2z = 6 and a line L: x/2 = (y - lambda)/(-1) = z/2. The plane and line intersect at a point A that lies on the xy-plane. Which of the following statements is/are correct?

1. The acute angle between line L and plane P is sin⁻¹(1/9)
2. Line L is perpendicular to plane P
3. Coordinates of A are (0, 6, 0)
4. Coordinates of A are (6, 6, 0)

Correct answer: Coordinates of A are (6, 6, 0)

Solution

Line L parametrically: x = 2t, y = lambda - t, z = 2t. Since A lies on xy-plane, z = 0 -> t = 0. So A = (0, lambda, 0). Substituting A into plane equation: 2(0) + lambda - 2(0) = 6 -> lambda = 6. Thus A = (0, 6, 0). For the angle between line and plane: direction vector of line d = (2, -1, 2), normal to plane n = (2, 1, -2). sin(theta) = |d.n|/(|d||n|) = |4 - 1 - 4|/(3*3) = 1/9. Acute angle = sin⁻¹(1/9). Both options (A) and (C) appear correct. But option (C) says coordinates are (0, 6, 0) and option (D) says (6, 6, 0). The correct intersection point is (0, 6, 0), matching option (C).

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