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Question
Find the equation of two straight lines which are parallel to x + 7y + 2 = 0 and at unit distance from the point (1, −1).
Answer 3:
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Solution
The equation of given line is x + 7y + 2 = 0 ... (1)
The equation of a line parallel to line x + 7y + 2 = 0 is given below:
\[x + 7y + \lambda = 0\] ... (2)
The line \[x + 7y + \lambda = 0\] is at a unit distance from the point (1, −1).
\[\therefore 1 = \left| \frac{1 - 7 + \lambda}{\sqrt{1 + 49}} \right|\]
\[ \Rightarrow \lambda - 6 = \pm 5\sqrt{2}\]
\[ \Rightarrow \lambda = 6 + 5\sqrt{2}, 6 - 5\sqrt{2}\]
Required lines :
\[x + 7y + 6 + 5\sqrt{2} = 0 \] \[\text{ and }x + 7y + 6 - 5\sqrt{2} = 0\]
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