COMPASSLearning

Talk to us Register Login

JEE Main · 2026 · 08 Apr 2026 · Shift 2 · Three Dimensional Geometry

Let a line

L_{1}

pass through the origin and be perpendicular to the lines

L_{2} : r = (3 + t) \hat{i} + (2 t - 1) \hat{j} + (2 t + 4) \hat{k}

and

L_{3} : r = (3 + 2 s) \hat{i} + (3 + 2 s) \hat{j} + (2 + s) \hat{k}, t

,

s \in R

. If

(a, b, c), a \in Z

, is the point on

L_{3}

at a distance of

17

from the point of intersection of

L_{1}

and

L_{2}

, then

(a + b + c)^{2}

is equal to ____.

Answer: 4

Solution

The directions of

L_{2}

and

L_{3}

are

(1, 2, 2)

and

(2, 2, 1)

; their cross product

(1, 2, 2) \times (2, 2, 1) = (- 2, 3, - 2)

gives the direction of

L_{1}

through the origin. Matching

(2 ℓ, - 3 ℓ, 2 ℓ)

on

L_{1}

with

(3 + t, 2 t - 1, 2 t + 4)

on

L_{2}

yields

t = - 1, ℓ = 1

, so the intersection is

P (2, - 3, 2)

. A general point of

L_{3}

is

Q (3 + 2 s, 3 + 2 s, 2 + s)

, and

P Q^{2} = 17

gives

(1 + 2 s)^{2} + (6 + 2 s)^{2} + s^{2} = 17 ⟹ 9 s^{2} + 28 s + 20 = 0,

with roots

s = - 2

and

s = - \frac{10}{9}

. An integer first coordinate

a = 3 + 2 s

requires

s = - 2

, so

Q (- 1, - 1, 0)

and

(a + b + c)^{2} = (- 2)^{2} = 4.

JEE Main · 2026 · 08 Apr 2026 · Shift 2 · Integer answer

Practise more: all JEE Main PYQs on Three Dimensional Geometry →

More JEE Main questions on Three Dimensional Geometry