If the straight lines $x=1+s, y=-3-\lambda s, z=1+\lambda s$ and $x=\frac{t}{2}, y=1+t, z=2-t$ with parameters $s$ and $t$ respectively, are co-planar then $\lambda$ equals

JEE Main · 2004 · 24 Apr 2004 · Offline · Three Dimensional Geometry

If the straight lines

x = 1 + s, y = - 3 - λ s, z = 1 + λ s

and

x = \frac{t}{2}, y = 1 + t, z = 2 - t

with parameters

s

and

t

respectively, are co-planar then

λ

equals

(A)-2✓ Correct
(B)-1
(C) $- \frac{1}{2}$
(D)0

Answer: (A)

Solution

Line one passes through

(1, - 3, 1)

with direction

(1, - λ, λ)

, and line two through

(0, 1, 2)

with direction

(\frac{1}{2}, 1, - 1)

, or equivalently

(1, 2, - 2)

. Coplanarity requires the scalar triple product of the join

(1, - 4, - 1)

with the two directions to vanish:

111 - 4 - λ 2 - 1 λ - 2 = 0

. Expanding gives

1 (2 λ - 2 λ) + 4 (- 2 - λ) - (2 + λ) = - 10 - 5 λ

, so

- 10 - 5 λ = 0

and

λ = - 2

JEE Main · 2004 · 24 Apr 2004 · Offline · Single correct option

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