Question

Assertion (A) : A line through the points (4, 7, 8) and (2, 3, 4) is parallel to a line through the points (−1, −2, 1) and (1, 2, 5). Reason (R) : Lines r→ = a₁→ + λb₁→ and r→ = a₂→ + μb₂→ are parallel if b₁→ · b₂→ = 0.

Direction ratios of lines l₁ and l₂ are <12, – 3, 9> and <4, q, – p> respectively. The values of p and q for which l₁ and l₂ are parallel are respectively :

(a) If the lines (x – 3)/1 = (1 – y)/1 = (z + 2)/p and (2 – x)/3 = (y + 1)/5 = (z + 56)/(2p) are perpendicular to each other, then find the value(s) of p.

(b) Opposite sides of a square are along the lines : r→ = i^ + 2j^ – 4k^ + λ(2i^ + 3j^ + 6k^) r→ = 3i^ + 3j^ – 5k^ + μ(2i^ + 3j^ + 6k^) Find the area of the square if direction ratios of other pair of opposite sides of the square are given by <– 3, 6, p>. Also, find the value of p.