JEE Main 2020 · Mathematics · Three Dimensional Geometry · hard

Let Q be the image (mirror reflection) of the point P(1,2,3) in the line x/1=(y-1)/(2)=z/2. The distance of Q from the origin is

Single correct option

Let

Q

be the image (mirror reflection) of the point

P(1,2,3)

in the line

\dfrac{x}{1}=\dfrac{y-1}{2}=\dfrac{z}{2}

. The distance of

Q

from the origin is

Show answer and solution

Answer: (C) $3\sqrt2$

A general point of the line is

F(\lambda,\,1+2\lambda,\,2\lambda)

with direction

(1,2,2)

. The foot of the perpendicular from

P

satisfies

\vec{PF}\cdot(1,2,2)=0

(\lambda-1)+2(2\lambda-1)+2(2\lambda-3)=0\Rightarrow 9\lambda=9\Rightarrow\lambda=1,

F=(1,3,2)

. The image is

Q=2F-P=(1,4,1)

because

F

is the midpoint of

PQ

. Then

|OQ|=\sqrt{1+16+1}=3\sqrt2

. The distractor

\sqrt{14}

|OF|

, the foot instead of the image.

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