Distance Between Two Lines: Formula & Solved Questions

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Distance between two lines is referred to as how far the two lines are located from each other. It is commonly referred to as the shortest distance between two parallel lines or the perpendicular distance between two lines. There are several formulas in coordinate geometry to calculate the same, out of which distance formula is one of the most commonly used formulas. For two non-intersecting lines lying in the same plane, the shortest distance is defined as the distance that is the shortest of all the distances between two points lying on both lines.

Table of Contents

Distance Between Two Lines
Steps to Calculate Distance Between Two Lines
Distance Between Two Lines Formula
Distance Between Two Parallel Lines
Shortest Distance Between Two Parallel Lines
Things to Remember
Sample Questions

Key Terms: Parallel Lines, Intersecting Lines, Skew Lines, Perpendicular, Points, Coordinate Geometry, Slope, Distance

Distance Between Two Lines

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The distance between two lines can be measured with the help of the two points that are there on each of the lines. The minimum distance between any two points lying on the lines is basically the distance between two straight lines. We come across various different sets of lines such as parallel lines, intersecting lines, or skew lines.

For parallel lines, the distance between two parallel lines is the perpendicular distance from any point on one line to the other line.
For intersecting lines, the shortest distance between both the lines is eventually zero.
For skew lines, the distance between two skew lines is equal to the length of the perpendicular between the lines.

Distance Between Two Lines

Read More: Distance Between Two Points

Steps to Calculate Distance Between Two Lines

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The steps to calculate the distance between two lines are as follows:

In order to calculate the distance, we check whether the given equations of parallel lines are in slope-intercept form (i.e. y= mx + c) or not.
The slope value should be common for both lines if the equations of lines are given in the slope-intercept form.
After that, we find the value of the interception point (c1 and c2) and find the value of the slope for both the lines.
Replace the previous values in the slope-intercept equation to calculate the value of y.
Finally, we put all the values in the distance formula discussed below to find the distance between two lines.

Distance Between Two Lines Formula

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Below given is the formula for distance between two parallel lines. The formula for the distance if we have the slope-intercept form of the two lines as y = mx + c1 and y = mx + c2 is:

In the above formula, c1 is the constant of line l1 and c2 is the constant for line l2. Also, in this case, m represents the slope of the line.

Distance Between Two Lines Formula

Distance Between Two Lines Formula

The formula for the distance if the equations of the parallel lines are given in the ax +by +c1 = 0 and ax +by +c2 = 0, is as follows:

Distance Between Two Parallel Lines

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We have previously discussed that the shortest distance between the two parallel lines can be determined using the length of the perpendicular segment between the lines. And it does not matter which perpendicular line we are choosing, as long as two points are on the line.

Because of this, we can now easily calculate the distance between two parallel lines and the distance between a point and a line. The formula for distance between two parallel lines having the slope-intercept form of equations of the two lines as y = mx + c1 and y = mx + c2

Here, c1 is the constant of line l1 and c2 is the constant for line l2, and m represents the slope of the line. Also if the equations of the parallel lines are given in the form ax + by + c1 = 0 and ax + by + c2 = 0

Shortest Distance Between Skew Lines

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The term Skew lines refer to the lines that exist in the multidimensional system, where two lines are non-parallel but never intersect with each other. Such a situation is possible only in 3-dimensions or more.

(x–x1)/a1 = (y–y1)/b1 = (z–z1)/c1

(x–x2)/a2 = (y–y2)/b2 = (z–z2)/c2

Things to Remember

Distance between two lines is referred to as the shortest distance of parallel lines that can be cut from one point to another.
On a 2D plane, the distance of two parallel lines can be calculated by finding the perpendicular distance between the lines.
Formula for perpendicular distance: d = |C1 – C2| / √ (A2 + B2)
The slope of two lines must be equal in order for them to be parallel.
When the slope of the given lines is equal and they are parallel to each other, we can compare it with the standard form of parallel lines equations to get the distance.
The shortest distance between the two parallel lines can be calculated easily through the length of the perpendicular segment between the lines.

Sample Questions

Ques. Calculate the distance between two lines 3x + 4y = 9 and 6x + 8y = 15. (5 Marks)

Ans. The equations of lines are:

3x + 4y = 9….(i)

6x + 8y = 15 Or 3x + 4y = 15/2 ….(ii)

Now we check whether the given lines are parallel or not.

From (i),

4y = -3x + 9

y = (-¾) x + (9 / 4)

Here, slope = m1 = -¾

From (ii),

8y = - 6x + 15

y = (- 6 / 8) x + (15 / 8)

y = (- ¾) x + (15 / 8)

Here, slope = m2 = - ¾

So, the slope of the given lines is equal so they are parallel to each other.

Thus, by comparing with the standard form of parallel lines equations, we get:

Here we have A = 3, B = 4, C1 = - 9, C2 = - 15 / 2

d = |C1 – C2| / √ (A2 + B2)

= |- 9 + (15 / 2)| / √ (9 + 16)

= |- 18 + 15| / 2 √25

= |-3| / (2 × 5)

= 3 / 10

Thus, the distance between the given lines is 3/10 units.

Ques. Find the value of k if the distance between two parallel lines 5x – 12y + 2 = 0 and 5x – 12y + k = 0 is given by 5/13 units. (5 Marks)

Ans. The equations of lines are:

5x – 12y + 2 = 0….(i)

5x – 12y + k = 0 ….(ii)

Now we check whether the given lines are parallel or not.

From (i),

12y = 5x + 2

y = (5 / 12) x + (2 / 12)

Here, slope = m1 = 5 / 12

From (ii),

12y = 5x + k

y = (5 / 12) x + (k / 12)

Here, slope = m2 = 5/12

From this we can say that the slope of the given lines is equal so they are parallel to each other and by comparing with the standard form of parallel lines equations, we get:

A = 5, B = -12, C1 = 2, C2 = k

d = |C1 – C2| / √ (A2 + B2)

5 / 13 = |2 – k| / √ (25 + 144)

5 / 13 = |2 - k| / √ 169

5 / 13 = |2 – k| / 13

5 = 2 – k

k = 2 – 5

= -3

Ques. State the distance between two lines 5x + 3y + 6 = 0 and 5x + 3y – 6 = 0? Calculate this by using the distance between two lines formula. (3 Marks)

Ans. In order to find the distance between two lines the parameters are,

a = 5, b = 3, c1 = 6, & c2 = -6

Using distance between two lines formula,

d = |c2 − c1| / √ a2+b2

d = |c2 − c1| / a2 + b2

d = | −6 −6| / √ 52 + 32

d = |−6 −6| / 52 + 32

d = 12 / √34

Thus, the distance between the two lines is 12 / √34.

Ques. Calculate the distance between the lines 4x + 3y + 6= 0 and 4x + 3y – 3 = 0. (3 Marks)

Ans. Given A = 4, B = 3, C1 = 6 and C2 = -3

Thus, the distance = | (C1 - C2) | / √ (A2 + B2)

= | (6 – 3) | / √ (16 + 9)

= 9 / √ 25

= 9 / 5

Ques. State how we calculate the shortest distance between two given points. (3 Marks)

Ans. As we have read above, the shortest distance between two lines can be calculated by using the distance formula, the length of a straight line drawn from one line to a point on the other line will give the distance between them. In case the coordinates of the lines are given, the distance formula can be used to calculate this.

Ques. State the formula to find the shortest distance between two parallel lines. (1 Mark)

Ans. Let d be the distance between two parallel lines. Then, d = |c2 − c1| / √ a2+b2

Ques. Does parallel line mean no solution? (3 Marks)

Ans. We have studied that the property of parallel lines is that they never intersect each other, other than at infinity, they can not have any solutions. For such solutions of parallel lines do not exist, hence it is known that the parallel lines have no solution. Thus, the equations of the parallel lines are known as the inconsistent set of equations.

Ques. What do you mean by parallel lines? Give examples. (3 Marks)

Ans. Parallel lines refer to the lines with the same slope. In such lines, the distance between them will never change. Few examples of the parallel lines are: 5x + 3y + 6 = 0 and 5x + 3y – 6 = 0 are parallel lines. Also, y = 5x + 5, and y = 5x - 7 are the parallel lines. The slope of parallel lines given here is the same. i.e., m1 = m2 = 5.

As the property of parallel lines is that they never intersect each other, other than at infinity, they can not have any solutions. Solutions to parallel lines do not exist, hence it is known that the parallel lines have no solution. And the equations of the parallel lines are known as the inconsistent set of equations.

Chapter Related Topics
Slope-Intercept Form Formula	Properties of Parallel Lines	Cartesian Coordinate System
Point of Intersection Formula	Angle Between a Line and a Plane	Constructing of Copy of Line Segments
Coplanarity of Two Lines	Angle Between Two Lines	Introduction to Three-Dimensional Geometry

Mathematics Study Guides
NCERT Solutions For Class 6 to 12	Math Formula	Mathematics Notes
Difference between in Maths	Math MCQs	Maths Study Material

Distance Between Two Lines: Formula & Solved Questions

Distance Between Two Lines

Steps to Calculate Distance Between Two Lines

Distance Between Two Lines Formula

Distance Between Two Parallel Lines

Shortest Distance Between Skew Lines

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

3.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

4.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

5.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

6.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

Similar Mathematics Concepts

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Distance Between Two Lines: Formula & Solved Questions

Distance Between Two Lines

Steps to Calculate Distance Between Two Lines

Distance Between Two Lines Formula

Distance Between Two Parallel Lines

Shortest Distance Between Skew Lines

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

3.Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

4.Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

5.Evaluate:\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

6. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

Similar Mathematics Concepts

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1.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

3.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

4.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

5.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

6.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]