Here you will learn what is the equation of line parallel to a given line with examples.
Let’s begin –
Equation of Line Parallel to a Line
The equation of the line parallel to a given line ax + by + c = 0 is
ax + by + \(\lambda\),
where \(\lambda\) is a constant.
Proof :
Let m be the slope of the line ax + by + c = 0, Then,
m = -\(a\over b\)
Since the required line is parallel to the given line, the slope of the required line is also m.
Let \(c_1\) be the y-intercept of the required line. Then, its equation is
y = mx + \(c_1\)
y = -\(a\over b\)x + \(c_1\)
\(\implies\) ax + by – b\(c_1\) = 0
\(\implies\) ax + by + \(\lambda\) = 0, where \(\lambda\) = -b\(c_1\) = constant.
Note : To write a line parallel to any given line we keep the expression containing x and y same and simply replace the given constant by a new constant \(\lambda\). The value of \(\lambda\) can be determined by some given condition.
Example : Find the equation of line which is parallal to the line 3x – 2y + 5 = 0 and passes through the point (5, -6).
Solution : The line parallel to the line 3x – 2y + 5 = 0 is
3x – 3y + \(\lambda\) = 0 …………..(i)
This passes through (5, -6)
\(\therefore\) 3 \(\times\) 5 – 2 \(\times\) -6 + \(\lambda\) = 0
\(\implies\) \(\lambda\) = -27.
Putting \(\lambda\) = -27 in (i) we get,
3x – 3y – 27 = 0, which is the required equation of line.
