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JEE Main 20254 AprilMorning Shift
Exam
JEE Main
Year
2025
Date
4 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Three Dimensional Geometry

JEE Main 2025 · 4 April · Morning Shift · Mathematics · Three Dimensional Geometry

Let A and B be two distinct points on the line L: x-6 3 = y-7 2 =...

Let A and B be two distinct points on the line L:x−63=y−72=z−7−2\mathrm{L}: \frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-7}{2}=\frac{\mathrm{z}-7}{-2}. Both A and B are at a distance 2172 \sqrt{17} from the foot of perpendicular drawn from the point (1,2,3)(1,2,3) on the line L . If O is the origin, then OA→⋅OB→\overrightarrow{O A} \cdot \overrightarrow{O B} is equal to:

Options

  1. A49
  2. B47
  3. C21
  4. D62

Correct answer

B. 47

Step-by-step solution

PQ→⋅b→=0⇒3(3λ+5)+2(2λ+5)−2(−2λ+4)⇒17λ=−17⇒λ=−1Q(3,5,9)\begin{aligned}& \overrightarrow{\mathrm{PQ}} \cdot \overrightarrow{\mathrm{ b}}=0 \\ & \Rightarrow 3(3 \lambda+5)+2(2 \lambda+5)-2(-2 \lambda+4) \\ & \Rightarrow 17 \lambda=-17 \Rightarrow \lambda=-1 \\ & \mathrm{Q}(3,5,9)\end{aligned}Let A (3 \mu+6,2 \mu+7,-2 \mu+7) (3 \mu+3)^2+(2 \mu+2)^2+(-2 \mu-2)^2=68 \Rightarrow \mu^2+2 \mu-3=0 \mu=-3 \text{ or } \mu=1A (-3,1,13)andB(9,9,5)OA→⋅OB→=−27+9+65=47and B(9,9,5) \overrightarrow{\mathrm{OA}} \cdot \overrightarrow{\mathrm{OB}}=-27+9+65=47
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