JEE Main 2026 Jan 23 Shift 1, Mathematics Q20: Lines in Space

JEE Main2026Jan 23, Shift 1Mathematics

Let the direction cosines of two lines satisfy the equations : $4 ℓ + m - n = 0$ and $2 mn + 10 n ℓ + 3 ℓ m = 0$ . Then the cosine of the acute angle between these lines is :

A
$\frac{10}{38}$
B
$\frac{20}{3 38}$
C
$\frac{10}{7 38}$
D
$\frac{10}{3 38}$

Solution

Eliminate $n$ . From the linear equation, $n = 4 ℓ + m$ . Substitute into the second:

$2 m (4 ℓ + m) + 10 ℓ (4 ℓ + m) + 3 ℓ m = 0$

$\Rightarrow 40 ℓ^{2} + 21 ℓ m + 2 m^{2} = 0 \Rightarrow (8 ℓ + m) (5 ℓ + 2 m) = 0$

Case 1: $m = - 8 ℓ$ , then $n = 4 ℓ - 8 ℓ = - 4 ℓ$ . Direction ratios $⟨ 1, - 8, - 4 ⟩$ .

Case 2: $m = - \frac{5 ℓ}{2}$ , then $n = 4 ℓ - \frac{5 ℓ}{2} = \frac{3 ℓ}{2}$ . Direction ratios $⟨ 2, - 5, 3 ⟩$ .

Cosine of angle.

$cos θ = \frac{∣ ( 1 ) ( 2 ) + ( - 8 ) ( - 5 ) + ( - 4 ) ( 3 ) ∣}{1 + 64 + 16 4 + 25 + 9} = \frac{∣2 + 40 - 12∣}{81 \cdot 38} = \frac{30}{9 38} = \frac{10}{3 38}$

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JEE Main 2026 Jan 23 Shift 1, Mathematics Q20: Lines in Space

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