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Question
Find the equations of two straight lines which are parallel to the line 12x + 5y + 2 = 0 and at a unit distance from the point (1, −1)
Solution
The equation of the given line is
12x + 5y + 2 = 0 ......(1)
Equation of any line parallel to the line (1) is
12x + 5y + k = 0 ......(2)
Given that line (2) is at a unit distance from the point (1, –1)
∴ 1 = `+- (12(1) + 5( - 1) + "k")/sqrt(12^2 + 5^2)`
`+- sqrt(144 + 25)` = 12 – 5 + k
`+- sqrt(169)` = 7 + k
k = – 7 ± 13
k = – 7 + 13 or k = – 7 – 13
k = 6 or k = – 20
∴ The equation of the required lines are
12x + 5y + 6 = 0 and 12x + 5y – 20 = 0
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