JEE Main · 2022 · 27 Jun 2022 · Shift 1 · Three Dimensional Geometry

If two straight lines whose direction cosines are given by the relations

l + m - n = 0, 3 l^{2} + m^{2} + c n l = 0

are parallel, then the positive value of c is :

(A)6✓ Correct
(B)4
(C)3
(D)2

Answer: (A)

Solution

From

l + m - n = 0

, we have

n = l + m

. Substituting into

3 l^{2} + m^{2} + c n l = 0

3 l^{2} + m^{2} + c l (l + m) = 0 ⟹ (3 + c) l^{2} + c l m + m^{2} = 0

Dividing by

m^{2}

and letting

t = \frac{l}{m}

(3 + c) t^{2} + c t + 1 = 0

For the lines to be parallel, the two direction ratios must be proportional, meaning this quadratic must have equal roots. Thus discriminant

D = 0

c^{2} - 4 (3 + c) = 0 ⟹ c^{2} - 4 c - 12 = 0 ⟹ (c - 6) (c + 2) = 0

The positive value is

c = 6

JEE Main · 2022 · 27 Jun 2022 · Shift 1 · Single correct option

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