The foot of perpendicular of the point (2, 0, 5) on the line x + 1 2 = y - 1 5 = z + 1 - 1 is . Then, which of the following is not correct? [2023]

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Solution

Q.
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 $(α, β, γ)$ . Then, which of the following is not correct [2023]

1 $\frac{γ}{α} = \frac{5}{8}$

2 $\frac{β}{γ} = - 5$

3 $\frac{α β}{γ} = \frac{4}{15}$

4 $\frac{α}{β} = - 8$

Ans.
(2)

$D.r's of line A B < α - 2, β, γ - 5 >$

Since $A B$ is perpendicular to

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

$∴$ $2 (α - 2) + 5 β - 1 (γ - 5) = 0$

$\Rightarrow 2 α + 5 β - γ + 1 = 0$ ...(i)

Also, $α, β, γ$ lies on the given line.

$∴$ $\frac{α + 1}{2} = \frac{β - 1}{5} = \frac{γ + 1}{- 1}$ ...(ii)

From (ii), (taking the first two and last two terms)

$α = \frac{2 β - 7}{5}, γ = \frac{- β - 4}{5}$

Putting these in (i), we get $4 β - 14 + 25 β + β + 4 + 5 = 0$

$\Rightarrow β = \frac{1}{6}$

which gives $α = \frac{- 4}{3}$ and $γ = \frac{- 5}{6}$ $\Rightarrow \frac{γ}{α} = \frac{\frac{- 5}{6}}{\frac{- 4}{3}} = \frac{5}{8}$ ,

$\frac{β}{γ} = \frac{1}{6} \times \frac{- 6}{5} = \frac{- 1}{5} \Rightarrow \frac{α β}{γ} = \frac{4}{15}, \frac{α}{β} = - 8, \frac{β}{γ} \neq - 5$

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