JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Three Dimensional Geometry

If the lines

\frac{x - k}{1} = \frac{y - 2}{2} = \frac{z - 3}{3}

and

\frac{x + 1}{3} = \frac{y + 2}{2} = \frac{z + 3}{1}

are co-planar, then the value of

k

is ____

Answer: 1

Solution

The first line passes through

(k, 2, 3)

with direction

(1, 2, 3)

and the second through

(- 1, - 2, - 3)

with direction

(3, 2, 1)

. They are coplanar when the joining vector and the two directions have zero scalar triple product:

- 1 - k 13 - 4 22 - 6 31 = 0

Expanding,

(- 1 - k) (2 - 6) + 4 (1 - 9) - 6 (2 - 6) = 4 + 4 k - 32 + 24 = 4 k - 4

. Setting this to zero gives

k = 1

JEE Main · 2021 · 25 Jul 2021 · Shift 2 · Integer answer

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