JEE Main · 2017 · 02 Apr 2017 · Offline · Three Dimensional Geometry

If the image of the point

P (1, - 2, 3)

in the plane,

2 x + 3 y - 4 z + 22 = 0

measured parallel to the line,

\frac{x}{1} = \frac{y}{4} = \frac{z}{5}

is Q , then PQ is equal to :

(A) $65$
(B) $35$
(C) $242$ ✓ Correct
(D) $42$

Answer: (C)

Solution

Send a line from

P (1, - 2, 3)

parallel to

\frac{x}{1} = \frac{y}{4} = \frac{z}{5}

: a general point is

(1 + λ, - 2 + 4 λ, 3 + 5 λ)

. It meets the plane

2 x + 3 y - 4 z + 22 = 0

when

2 (1 + λ) + 3 (- 2 + 4 λ) - 4 (3 + 5 λ) + 22 = 0

, which simplifies to

6 - 6 λ = 0

, so

λ = 1

and the crossing point is

(2, 2, 8)

. That crossing is the midpoint of

P Q

, so

Q = (3, 6, 13)

and

P Q = 2^{2} + 8^{2} + 1 0^{2} = 168 = 242

JEE Main · 2017 · 02 Apr 2017 · Offline · Single correct option

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