Equation of Line Parallel to a Line

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.

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