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Question
The direction ratios of the line x − y + z − 5 = 0 = x − 3y − 6 are proportional to
Options
3, 1, −2
2, −4, 1
\[\frac{3}{\sqrt{14}}, \frac{1}{\sqrt{14}}, \frac{- 2}{\sqrt{14}}\]
\[\frac{2}{\sqrt{41}}, \frac{- 4}{\sqrt{41}}, \frac{1}{\sqrt{41}}\]
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Solution
3, 1, −2
We have ,
x − y + z − 5 = 0 = x − 3y − 6
\[\Rightarrow x - 3y - 6 = 0 \]
\[ x - y + z - 5 = 0\]
\[ \Rightarrow x = 3y + 6 . . . \left( 1 \right) \]
\[ x - y + z - 5 = 0 . . . \left( 2 \right)\]
From (1) and (2),
we get ,
\[3y + 6 - y + z - 5 = 0\]
\[ \Rightarrow 2y + z + 1 = 0\]
\[ \Rightarrow y = \frac{- z - 1}{2} \]
\[y = \frac{x - 6}{3} \left[\text { From } \left( 1 \right) \right]\]
\[ \therefore \frac{x - 6}{3} = y = \frac{- z - 1}{2}\]
So, the given equation can be re-written as
\[\frac{x - 6}{3} = \frac{y}{1} = \frac{z + 1}{- 2}\]
Hence, the direction ratios of the given line are proportional to 3, 1, -2 .
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