The foot of perpendicular of the point $(2,0,5)$ on the line $\frac{x+1}{2}=\frac{y-1}{5}=\frac{z+1}{-1}$ is $(\alpha, \beta, \gamma)$. Then. Which of the following is NOT correct?

(C) $\frac{\beta}{\gamma}=-5$

(A) $\frac{\alpha \beta}{\gamma}=\frac{4}{15}$

(B) $\frac{\alpha}{\beta}=-8$

(D) $\frac{\gamma}{\alpha}=\frac{5}{8}$

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Three Dimensional Geometry

The foot of perpendicular of the point

(2, 0, 5)

on the line

\frac{x + 1}{2} = \frac{y - 1}{5} = \frac{z + 1}{- 1}

(α, β, γ)

. Then.
Which of the following is NOT correct?

Answer: (C)

Write a general point of the line as

(2 λ - 1, 5 λ + 1, - λ - 1)

. The vector from it to

(2, 0, 5)

must be perpendicular to the direction

(2, 5, - 1)

2 (3 - 2 λ) - 5 (5 λ + 1) - (6 + λ) = 0

, giving

λ = - \frac{1}{6}

. The foot is

(α, β, γ) = (- \frac{4}{3}, \frac{1}{6}, - \frac{5}{6})

. Then

\frac{α β}{γ} = \frac{4}{15}

\frac{α}{β} = - 8

and

\frac{γ}{α} = \frac{5}{8}

all hold, while

\frac{β}{γ} = \frac{1/6}{- 5/6} = - \frac{1}{5}

. The relation that does not hold is

\frac{β}{γ} = - 5

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Single correct option