JEE Main · 2013 · 07 Apr 2013 · Offline · Three Dimensional Geometry

If the lines

\frac{x - 2}{1} = \frac{y - 3}{1} = \frac{z - 4}{- k}

and

\frac{x - 1}{k} = \frac{y - 4}{2} = \frac{z - 5}{1}

are coplanar, then

k

can have

(A)exactly one value
(B)exactly two values✓ Correct
(C)exactly three values
(D)any value

Answer: (B)

Solution

The lines are coplanar when the determinant built from their direction ratios and the joining vector vanishes. With directions

(1, 1, - k)

and

(k, 2, 1)

and joining vector

(- 1, 1, 1)

, the condition

- 1 1 k 112 1 - k 1 = 0

expands to

- k^{2} - 3 k = 0

, so

k (k + 3) = 0

and

k = 0

k = - 3

, exactly two values.

JEE Main · 2013 · 07 Apr 2013 · Offline · Single correct option

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