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.