Example 1 :
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 :
Since the required line is parallel to the line 12x + 5y + 2 = 0.
Equation of the required line will be in the form
12x + 5y + k = 0
Distance between the point (1, -1) and 12x + 5y + k = 0 is 1.
Distance between the point and a line
= |Ax + By + C|/√A^{2} + B^{2}
= |12(1) + 5(-1) + k|/√12^{2} + 5^{2}
1 = |7 + k|/√169
1 = |7 + k|/13
13 = 7 + k k = 13 - 7 = 6 |
-13 = 7 + k k = -13 - 7 = -20 |
Hence the required lines are 12x + 5y + 6 = 0 and 12x + 5y - 20 = 0.
Example 2 :
Find the equations of straight lines which are perpendicular to the line 3x+ 4y −6 = 0 and are at a distance of 4 units from (2, 1).
Solution :
Since the required line is perpendicular to the line 3x+ 4y −6 = 0
Equation of the required line will be in the form
4x - 3y + k = 0
Distance between the point (2, 1) and 4x - 3y + k = 0 is 4.
Distance between the point and a line
= |Ax + By + C|/√A^{2} + B^{2}
4 = |4(2) - 3(1) + k|/√4^{2} + (-3)^{2}
4 = |8 - 3 + k|/√16 + 9
4 = |5 + k|/5
20 = |5 + k|
20 = 5 + k k = 20 - 5 = 15 |
-20 = 5 + k k = -20 - 5 = -25 |
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