Distance between 2 parallel lines is the perpendicular distance from any point to one of the lines. In this article you will learn parallel lines definition , how to find distance between them and solved examples.
The graph of the function, a pair of railroad tracks or the opposite sides of a parallelogram or keys of a piano can be a few examples of parallel lines. A common property which can be found in the above examples is that the two railroad tracks never meet, the opposite sides of a parallelogram will never intersect or the piano keys are parallel to each other. This article explains the method of finding the distance between two parallel lines with appropriate examples.
Parallel Lines Definition
Parallel lines are those lines which never meet each other. When the distance between a pair of lines is the same throughout, it can be called parallel lines. It is denoted by “||”. The main criteria for any two lines to be parallel are that they have to be drawn on the same plane. They are always equidistant from each other.
The lines can be extended till infinity. The slopes of two parallel lines are equal.
How to Find the Distance Between Two Parallel Lines
The method for calculating the distance between two parallel lines is as follows:
- Ensure whether the equations of the given parallel lines are in slope-intercept form (y=mx+c).
- The point of interception (c_{1} and c_{2}) and slope value which is common for both the lines has to be determined.
- After obtaining the above values, substitute them in the slope-intercept equation to find y.
- Finally, put all the above values in the distance formula to find the distance between two parallel lines.
The two parallel lines can be taken in the form
y = mx + c_{1} … (i)
and y = mx + c_{2 } … (ii)
The line (i) will intersect the x-axis at point A (–c_{1}/m, 0) as shown in the figure.
The length of the perpendicular from point A to the line (ii) is of the same length as the distance between two lines.
Therefore, the distance between the lines (i) and (ii) is
|(–m)(–c_{1}/m) + (–c_{2})|/√(1 + m^{2}) or d = |c_{1}–c_{2}|/√(1+m^{2}).
Distance d between two parallel lines y = mx + c_{1} and y = mx + c_{2} is given by
d = |C_{1}–C_{2}|/√A^{2} + B^{2 } |
Distance formula d = |a_{1}x_{1}+b_{1}y_{1}+c_{1}| / √a_{1}^{2}+b_{1}^{2} , where d is the distance between two parallel lines. x1 and y1 are the two intersection points of the lines with the axis in a cartesian plane, while a1 and b1 are the coefficients of variable x and y of the line.
The equation of line through which equation of the distance formula is written as
a_{1}x+b_{1}y+c_{1} = 0
Considering the following equations of 2 parallel lines, we can calculate the distance between those lines using the distance formula
ax+by+c = 0
ax+by+c_{1} = 0
Using above 2 equations we can conclude that
Distance formula d = |c-c_{1}| / √a^{2}+b^{2}
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Solved Examples
Example 1: Find the distance of the point (4, –6) from the line 2x – 7y – 24 = 0.
Solution:
Given line is 2x – 7y – 24 = 0. …… (1)
Comparing (1) with general equation of line Ax + By + C = 0, we get
A = 2, B = –7 and C = –24.
Given point is (x_{1}, y_{1}) = (4, –6).
The distance of the given point from given line is d = |Ax_{1 }+ By_{1} + C|/√A^{2}+B^{2} = 26/7.2 = 3.571
Example 2: What is the distance of the lines 2x − 3y = 4 from the point (1, 1) measured parallel to the line x+y=1?
Solution:
The slope of line x + y = 1 is -1. It makes an angle of 135^{∘} with the x-axis.
The equation of the line passing through (1,1) and making an angle of 135^{∘} is,
x − 1/ cos135^{o }= y − 1/sin135^{o }= r
x − 1/(- 1/√2) = y − 1/1/√2 = r
Co-ordinates of any point on this line are
(1 − r/√2, 1 + r/√2)
If this point lies on 2x − 3y = 4, then
2(1 − r/√2) − 3(1 + r/√2) = 4
r = √2.
Example 3: Estimate the distance between the two parallel lines y=2x+7 and y=2x+5.
Solution:
The distance between two parallel lines with the same slope ( i.e. m_{1}=m_{2}) is given by |c_{1}-c_{2}|.
Here, the equations of parallel lines are y = 2x + 7 and y = 2x + 5.
Slopes are same m_{1} = m_{2} = 2 and c_{1} = 7 ,c_{2} = 5.
So, the distance between two parallel lines is given by
=> |c_{1}-c_{2}| = |7-5| = 2
Example 4: Calculate the distance between the parallel lines 3x+4y+7=0 and 3x+4y−5=0 from origin.
Solution:
d_{1}= distance of perpendicular from (0,0) to 3x+4y+7 = 0
Using formula, we get
d_{1}= (3 × 0 + 4 × 0 + 7) /√(3^{2}+4^{2}) = 7/5
d_{2 }= (3 × 0 + 4 × 0 + (−5) )/ √(3^{2}+4^{2}) = −5/5
Required distance is =|d1 − d2| = ∣7/5 − (−5/5)∣ = 12/5. Answer!!
Example 5: Find the distance from the line 6x – 4y + 36 = 0 to point (0, 0).
Solution:
Writing the given equation in the normal form, we get
x – 2/3 y + 6 = 0
Substituting x = 0, y = 0 in it,
d = 1/1 (0) – 2/3 (0) + 6 = 6
Thus, the required distance = 6.
Example 5: Let PS be the median of the triangle with vertices
A) 2x − 9y − 7 = 0
B) 2x − 9y − 11 = 0
C) 2x + 9y − 11 = 0
D) 2x + 9y + 7 = 0
Solution:
S = midpoint of
The required equation is
Example 6: The graph of the function
A) A straight line passing through (0,−sin^2 [1]) with slope 2
B) A straight line passing through (0, 0)
C) A parabola with vertex 75 degrees
D) A straight line passing through the point (π/ 2, −sin^2 [1]) and parallel to the x-axis
Solution:
Example 7: The line 3x + 2y = 24 meets y-axis at A and x-axis at B. The perpendicular bisector of AB meets the line through (0,−1) parallel to the x-axis at C. The area of the triangle ABC is
A) 182sq.units
B) 91sq.units
C) 48sq.units
D) None of these
Solution:
The coordinates of A and B are (0, 12) and (8, 0) respectively. The equation of the perpendicular bisector of AB is
Equation of a line passing through (0, 1) and parallel to the x-axis is y=−1.
This meets (i) at C, Therefore the coordinates of C are
Hence the area of the triangle ABC is
Example 8: The diagonals of a parallelogram PQRSare along the lines x + 3y = 4 and 6x − 2y = 7. Then PQRS must be a
A) Rectangle
B) Square
C) Cyclic quadrilateral
D) Rhombus
Solution:
m1 = −1/3 and m2 = 3.
Hence lines x + 3y = 4 and 6x − 2y = 7 are perpendicular to each other.
Therefore the parallelogram is a rhombus.
Example 9:
Find the distance between the lines 4x +3y+6= 0 and 4x+3y-3= 0.
Solution:
Here A = 4, B = 3, C1 = 6 and C2 = -3
So the distance =| (C1-C2) | / √(A^{2}+B^{2})
=| (6–3) | / √(16+9) = 9/√25 = 9/5
Example 10: A straight line through the origin O meets the parallel lines 4x + 2y = 9 and 2x + y + 6 = 0 at points P and Q respectively. Then, the point O divides the segment PQ in the ratio
(a) 1:2
(b) 3:4
(c) 2:1
(d) 4:3
Solution:
Given lines are 4x+2y = 9 and 2x+y+6 = 0.
The distance of the origin from 4x+2y-9 = 0 is |-9|/√42+22 = 9/√20.
Also the distance of origin from 2x+ y + 6 = 0 is |6|/√(22 +12 ) = 6/√5.
So, the required ratio is (9/√20) / (6/√5) = 3/4.
Hence option B is the answer.