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:

Answer 1

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 :

\[x + 7y + 6 + 5\sqrt{2} = 0 \] \[\text{ and }x + 7y + 6 - 5\sqrt{2} = 0\]