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JEE Main 2023 · 25 Jan · Shift 2

JEE Main 2023 Mathematics question on Three Dimensional Geometry

The foot of perpendicular of the point (2, 0, 5) on the line x+12=y−15=z+1−1\dfrac{x+1}{2} = \dfrac{y-1}{5} = \dfrac{z+1}{-1} is (α,β,γ)(\alpha, \beta, \gamma). Then, which of the following is NOT correct?
  1. A.
    αβγ=415\dfrac{\alpha\beta}{\gamma} = \dfrac{4}{15}
  2. B.
    γα=58\dfrac{\gamma}{\alpha} = \dfrac{5}{8}
  3. C.
    βγ=−5\dfrac{\beta}{\gamma} = -5
  4. D.
    αβ=−8\dfrac{\alpha}{\beta} = -8
Show answer and solution

Correct option: C

Solution

A point on the line is (-1+2t, 1+5t, -1-t). The vector from this point to (2,0,5) must be perpendicular to direction (2,5,-1).

Vector: (3-2t, -1-5t, 6+t). Dot with (2,5,-1):
2(3-2t) + 5(-1-5t) + (-1)(6+t) = 0
6-4t - 5-25t - 6-t = 0
-5 - 30t = 0 → t = -1/6

So α = -1 + 2(-1/6) = -1 - 1/3 = -4/3
β = 1 + 5(-1/6) = 1 - 5/6 = 1/6
γ = -1 - (-1/6) = -1 + 1/6 = -5/6

Check option 3: β/γ = (1/6)/(-5/6) = -1/5, NOT -5. So option 3 is NOT correct.

[JEE Main 2023 · 25 Jan · Shift 2]

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