JEE Main · 2024 · 09 Apr 2024 · Shift 2 · Three Dimensional Geometry

The square of the distance of the image of the point (6, 1, 5) in the line

\frac{x - 1}{3} = \frac{y}{2} = \frac{z - 2}{4}

, from the origin is ____.

Answer: 62

Solution

Line:

\frac{x - 1}{3} = \frac{y}{2} = \frac{z - 2}{4} = λ

, general point

M = (3 λ + 1, 2 λ, 4 λ + 2)

M

is the midpoint of

A (6, 1, 5)

and its image, so

A M ⊥ (3, 2, 4)

A M = (3 λ - 5, 2 λ - 1, 4 λ - 3)

3 (3 λ - 5) + 2 (2 λ - 1) + 4 (4 λ - 3) = 0 ⟹ 29 λ - 29 = 0 ⟹ λ = 1

M = (4, 2, 6)

. The image

I = 2 M - A = (8 - 6, 4 - 1, 12 - 5) = (2, 3, 7)

O I^{2} = 2^{2} + 3^{2} + 7^{2} = 4 + 9 + 49 = 62

JEE Main · 2024 · 09 Apr 2024 · Shift 2 · Integer answer

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