Question

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 $(a, \beta, \gamma)$. Then which of the following is NOT correct?

• A. $\frac{\gamma}{\alpha}=\frac{5}{8}$
• B. $\frac{\beta}{\gamma}=-5$
• C. $\frac{\alpha \beta}{\gamma}=\frac{4}{15}$
• D. $\frac{\alpha}{\beta}=-8$

Answer 1

Answer: (B)

$\frac{\beta}{\gamma}=-5$

Answer 2

The correct answer is (B) : $\frac{\beta}{\gamma}=-5$
L:2x+1​=5y−1​=−1z+1​=λ

Let foot of perpendicular is
P(2λ−1,5λ+1,−λ−1)
PA=(3−2λ)i^−(5λ+1)j^​+(6+λ)k^
Direction ratio of line ⇒b=2i^+5j^​−k^
Now, ⇒PA⋅b=0
⇒2(3−2λ)−5(5λ+1)−(6+λ)=0
⇒λ=6−1​
P(2λ−1,5λ+1,−λ−1)≡P(α,β,γ)
⇒α=2(−61​)−1=−34​⇒α=−34​
⇒β=5(−61​)+1=61​⇒β=61​
⇒γ=−λ−1=61​−1⇒γ=−65​