JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Three Dimensional Geometry

Let a line L pass through the point P(2,3,1) and be parallel to the line

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

. If the distance of L from the point

(5, 3, 8)

α

, then

3 α^{2}

is equal to ____.

Answer: 158

Solution

The line

L

is parallel to the intersection of the planes whose normals are

(1, 3, - 2)

and

(1, - 1, 2)

, so its direction is their cross product

(4, - 4, - 4) ∥ (1, - 1, - 1)

. For the distance of the point

Q (5, 3, 8)

from

L

through

P (2, 3, 1)

, with

P Q = (3, 0, 7)

and

d = (1, - 1, - 1)

P Q \times d = (7, 10, - 3)

, so

α^{2} = \frac{∣ P Q \times d ∣ ^{2}}{∣ d ∣ ^{2}} = \frac{49 + 100 + 9}{3} = \frac{158}{3}

. Therefore

3 α^{2} = 158

JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Integer answer

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