JEE Main 2026 — Jan 21, Shift 1 — Mathematics solution

JEE Main2026Jan 21, Shift 1Mathematics

Let $(α, β, γ)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5, 4, 2)$ on the line $r = (- \hat{i} + 3 \hat{j} + \hat{k}) + λ (2 \hat{i} + 3 \hat{j} - \hat{k})$ .

Then the length of the projection of the vector $α \hat{i} + β \hat{j} + γ \hat{k}$ on the vector $6 \hat{i} + 2 \hat{j} + 3 \hat{k}$ is :

A
$\frac{15}{7}$
B
4
C
$\frac{18}{7}$
D
3

Solution

A general point on the line is $P (λ) = (2 λ - 1, 3 λ + 3, - λ + 1)$ with direction $d = (2, 3, - 1)$ .

$A P = P - (5, 4, 2) = (2 λ - 6, 3 λ - 1, - λ - 1)$ . Perpendicularity $A P \cdot d = 0$ gives $4 λ - 12 + 9 λ - 3 + λ + 1 = 0 \Rightarrow λ = 1$ .

So $(α, β, γ) = (1, 6, 0)$ . Projection on $(6, 2, 3)$ : $\frac{∣6 + 12 + 0∣}{36 + 4 + 9} = \frac{18}{7}$ .

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