Distance between parallel lines MCQ Quiz - Objective Question with Answer for Distance between parallel lines - Download Free PDF

Answer : 4

Solution :

Given equation of pair of lines is

2x² + 4xy - py² + 4x + qy + 1 = 0

Comparing with

ax² + 2hxy + by² + 2gx + 2fy + c = 0, we get a = 2, h = 2, b = -p

If the given equation represents a pair of straight lines, then

h² ≥ ab

⇒ 4 ≥ -2p

⇒ 2 ≥ -p

⇒ p ≥ -2

∴ Possible values of p from domain [-5, 5] are -2, -1, 0, 1, 2, 3, 4, 5.

∴ Number of integral values of p = 8

Calculation:

Let the two parallel lines be ax + by + c₁ = 0 and ax + by + c₂ = 0

∴ (ax + by + c1)(ax + by + c2) = 0

⇒ a²x² + 2abxy + b²y² + a(c₁ ​+ c_2​)x + b(c₁ ​+ c_2​)y + c_1​c₂ ​= 0 

Comparing with x2 + 4xy + py2 + 3x + qy - 4 = 0, we get:

a² = 1, ab = 2, b² = p, a(c1 ​+ c2​) = 3, b(c1 ​+ c2​) = q and c1c2​ = - 4

Solving them we get:

a = 1, b = 2, p = 4, q = 6, (c₁, c₂) = (4, - 1)

∴ The parallel lines are x + 2y + 4 = 0 and x + 2y - 1 = 0

∴ Distance between the lines =  = √5

⇒ λ = √5

⇒ λ² = 5

∴ The value of λ² is 5.

The correct answer is Option 1.

Concept:

Shortest distance between two lines  and  is given by

Calculation:

Given lines L₁ : 

L1 can be written as in vector from as 

Also, L₂ : 

L₂ can be written as in vector from as 

Here, ,

,

,

Now, 

⇒ 

Now, 

So,  = 4[4 + 3 + 42] = 196

And 

∴ Shortest distance between given lines is

= 14

∴ The shortest distance is 14.

Given:

The parallel lines y - 8 = 0 and y + 1 = 0.

Formula used:

Line 1: ax + by = c₁

Line 2: ax + by = c2

Where lines 1 and line 2 are parallel to each other.

d = 

d, is the distance between two parallel lines.

Calculation:

Line 1: 0x + y = 8

Line 2: 0x + y = -1

Now, on comparing the equation of the lines with its standard form, we get,

a = 0, b = 1, c₁ = 8, c₂ = -1

Where lines 1 and line 2 are parallel to each other.

Therefore, 

⇒d = 

⇒ d = 

⇒ d = 9

∴ The distance between two parallel lines is 9.

Concept:

Distance between lines  and  is given by d = 

Calculation:

Given line,  =  =  

Comparing with , we get:

x1 = 2, y1 = 1, z1 = -1 and a1 = 1, b1 = 2, c1 = 2

∴  ⇔ 

and,  ⇔ 

Now, and equation of z-axis is   =  =  

Comparing with , we get:

x₂ = 0, y₂ = 0, z₂ = 0 and a₂ = 0, b₂ = 0, c₂ = 1

∴  ⇔ 

and,  ⇔ 

Now, 

 =  = 

Maginutude of  =  =  = 

Also,  =  = -4 + 1 = -3

∴ Distance, d = 

= 

= 

∴ The shortest distance between the z-axis and the line  =  =   is 

The correct answer is Option 2.

Concept:

Distance between parallel lines ax + by + c1  and ax + by + c2 is given by 

Calculation:

Here, given straight lines are 6x - 4y = 5 and 3x - 2y = 4

Multiply, 3x - 2y = 4 by 2 we get, 

6x - 4y = 8 

⇒ 6x - 4y - 8 = 0

So, distance between 6x - 4y - 5 = 0 and 6x - 4y - 8 = 0 is given by, 

d = 

Hence, option (4) is correct.

CONCEPT:

The distance between the parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is given by:

CALCULATION:

Here, we have to find the distance between the parallel lines 4x - 3y + 5 = 0 and 4x - 3y + 7 = 0.

By comparing the equations of the given line with x + by + c1 = 0 and ax + by + c2 = 0 we get

⇒ a = 4, b = - 3, c1 = 5 and c2 = 7.

As we know that, the distance between the parallel lines is given by: 

⇒ 

So, the distance between the given parallel lines is 2/5

Hence, option C is the correct answer.

Concept:

Condition for parallel lines: If the two lines are parallel, then their general forms of equations will differ only in the constant term and they will have the same coefficients of x and y. 

That is, 

ax + by + c1 = 0      ----(a)

ax + by + c2 = 0      ----(b)

Distance between two parallel lines: Distance d between two parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is given by,

      ----(A)

Calculation:

We have, 

 x + 2y = 5      ----(1)

Comparing it with the standard equation of a straight line i.e., ax + by + c = 0, we get

a₁ = 1, b₁ = 2, c₁ = -5

And, 2x + 4y = 11 

⇒ x + 2y = 11/2      ----(2)

Comparing it with the standard equation of a straight line i.e., ax + by + c = 0, we get

a₂ = 1, b₂ = 2, c₂ = -11/2

Thus, the coefficients of x and y in equations (1) & (2) are the same.

That is, (a1 = a2) and (b1 = b2)

Thus, these are parallel lines.

Now, we have a = 1, b = 2, c₁ = -5, and c₂ = -11/2

      [using (A)]

Hence, distance between the straight lines x + 2y = 5 and 2x + 4y = 11 is .

Concept:

Distance between two parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is 

Calculation:

We have, 3x – 4y + 12 = 0 and 3x – 4y = 6⇒ 3x – 4y - 6 = 0

∴ Distance between them = 

= 18/5

Let, equation of line midway between the given lines 3x - 4y + c = 0

As, it is in the middle, distance from line 3x – 4y + 12 = 0 will be 

Now, 9/5 =

⇒ 12 - c = 9

⇒ c = 3

Hence, option (4) is correct.

Given:

The parallel lines y - 8 = 0 and y + 1 = 0.

Formula used:

Line 1: ax + by = c₁

Line 2: ax + by = c2

Where lines 1 and line 2 are parallel to each other.

d = 

d, is the distance between two parallel lines.

Calculation:

Line 1: 0x + y = 8

Line 2: 0x + y = -1

Now, on comparing the equation of the lines with its standard form, we get,

a = 0, b = 1, c₁ = 8, c₂ = -1

Where lines 1 and line 2 are parallel to each other.

Therefore, 

⇒d = 

⇒ d = 

⇒ d = 9

∴ The distance between two parallel lines is 9.

Concept:

Distance between parallel lines ax + by + c1  and ax + by + c2 is given by 

Calculation:

Here, given straight lines are 6x - 4y = 5 and 3x - 2y = 4

Multiply, 3x - 2y = 4 by 2 we get, 

6x - 4y = 8 

⇒ 6x - 4y - 8 = 0

So, distance between 6x - 4y - 5 = 0 and 6x - 4y - 8 = 0 is given by, 

d = 

Hence, option (4) is correct.

CONCEPT:

The distance between the parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is given by:

CALCULATION:

Here, we have to find the distance between the parallel lines 4x - 3y + 5 = 0 and 4x - 3y + 7 = 0.

By comparing the equations of the given line with x + by + c1 = 0 and ax + by + c2 = 0 we get

⇒ a = 4, b = - 3, c1 = 5 and c2 = 7.

As we know that, the distance between the parallel lines is given by: 

⇒ 

So, the distance between the given parallel lines is 2/5

Hence, option C is the correct answer.

Concept:

Condition for parallel lines: If the two lines are parallel, then their general forms of equations will differ only in the constant term and they will have the same coefficients of x and y. 

That is, 

ax + by + c1 = 0      ----(a)

ax + by + c2 = 0      ----(b)

Distance between two parallel lines: Distance d between two parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is given by,

      ----(A)

Calculation:

We have, 

 x + 2y = 5      ----(1)

Comparing it with the standard equation of a straight line i.e., ax + by + c = 0, we get

a₁ = 1, b₁ = 2, c₁ = -5

And, 2x + 4y = 11 

⇒ x + 2y = 11/2      ----(2)

Comparing it with the standard equation of a straight line i.e., ax + by + c = 0, we get

a₂ = 1, b₂ = 2, c₂ = -11/2

Thus, the coefficients of x and y in equations (1) & (2) are the same.

That is, (a1 = a2) and (b1 = b2)

Thus, these are parallel lines.

Now, we have a = 1, b = 2, c₁ = -5, and c₂ = -11/2

      [using (A)]

Hence, distance between the straight lines x + 2y = 5 and 2x + 4y = 11 is .

Concept:

Distance between lines  and  is given by d = 

Calculation:

Given line,  =  =  

Comparing with , we get:

x1 = 2, y1 = 1, z1 = -1 and a1 = 1, b1 = 2, c1 = 2

∴  ⇔ 

and,  ⇔ 

Now, and equation of z-axis is   =  =  

Comparing with , we get:

x₂ = 0, y₂ = 0, z₂ = 0 and a₂ = 0, b₂ = 0, c₂ = 1

∴  ⇔ 

and,  ⇔ 

Now, 

 =  = 

Maginutude of  =  =  = 

Also,  =  = -4 + 1 = -3

∴ Distance, d = 

= 

= 

∴ The shortest distance between the z-axis and the line  =  =   is 

The correct answer is Option 2.

Concept:

Distance between two parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is 

Calculation:

We have, 3x – 4y + 12 = 0 and 3x – 4y = 6⇒ 3x – 4y - 6 = 0

∴ Distance between them = 

= 18/5

Let, equation of line midway between the given lines 3x - 4y + c = 0

As, it is in the middle, distance from line 3x – 4y + 12 = 0 will be 

Now, 9/5 =

⇒ 12 - c = 9

⇒ c = 3

Hence, option (4) is correct.