Two lines are said to be parallel if the angle between them is zero. There is a result on parallel line which is states as : “Two lines are parallel if and only if their slopes are equal.”
i.e. If the slopes of two lines which are parallel are m1 and m2 then m1 = m2.
Example:
The slope of a line is 3/2. If a line is parallel to this line, what is the slope of this new line?
Solution:
By the condition for the slopes of parallel lines,
m1=m2.
So, the slope of new line = 3/2.
Finding the Equation of Parallel Lines:
For finding the equation of a line parallel to ax+by+c=0 and passing through the point (x1,y1), there are 2 methods.
Method 1:
(1) Finding the slope of ax+by+c=0 which can be obtained by the expression –a/b.
i.e. slope of ax+by+c=0 is m=-a/b.
(2) Then the corresponding equation of parallel line can be found using point slope form.
i.e., the required equation is,
y-y1=m(x-x1)
Example:
Find the equation of line parallel to 3x+y+3=0 and passing through (-1,2).
Solution:
The slope of given line, m = -3/1=-3
(x1,y1) = (-1,2).
The equation of parallel line is,
y-y1=m(x-x1)
y-2 = -3 (x+1)
y-2 = -3x-3
3x+y+1=0.
Method 2:
(1) Take the equation of line parallel to ax+by+c=0 as ax+by+k=0 where k is a constant.
(2) The value of ‘k’ can be found by substituting the given point (x1,y1) in the equation ax+by+k=0.
(3) Substitute ‘k’ back into the equation ax+by+k=0 which is the required equation.
Example:
Find the equation of line parallel to 3x+y+3=0 and passing through (-1,2).
Solution:
The equation of line parallel to 3x+y+3=0 is,
3x+y+k=0.
Here x=-1 and y=2
3(-1)+2+k=0
k-1=0
k=1
So the required equation of parallel line is,
3x+y+1=0.
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