The equations of the plane π, and the lines L1 and l2 are given by

π: tx-2y+z=13

L1: r = (2i + j + 3k) + λ [2t i + (t² – 1) j – 2k]

L2: x – 4 = y, z = 5

where t and λ are real constants.

(a) Given that the shortest distance from the point P with coordinates (3, 3, -5) to π is 6, find the possible values of t.

(b) Show that L1 is parallel to π, and find a condition on t such that L1 is not on π.

For the case where t=2,

(i) show that L2 lies on π

(ii) given that A is a point on L1, and B is a point on L2, find the position vectors of A and B such that AB is perpendicular to both L1 and L2.

(iii) find the vector equation of the line of reflection of L1 in π.

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