Explanation
Solving for Foot of Perpendicular
Let the point P(α,β,γ) on the line be represented in terms of a parameter λ:
From this, we get the coordinates:
The direction ratios (DRs) of the line joining the given point (2,0,5) and the foot P(α,β,γ) are:
(2λ−1−2,5λ+1−0,−λ−1−5)=(2λ−3,5λ+1,−λ−6)
Since this line is perpendicular to the given line, the dot product of their DRs must be zero:
2(2λ−3)+5(5λ+1)−1(−λ−6)=0
Now, substitute λ=−61 to find α,β, and γ:
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α=2(−61)−1=−34=−68
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β=5(−61)+1=61
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γ=−(−61)−1=−65
Checking Option (1)
The first option states that γβ=−5. Let's calculate the actual ratio:
Since −51=−5, Option (1) is NOT correct.
Correct Answer: (1)
