Shortest Distance MCQ Quiz - Objective Question with Answer for Shortest Distance - Download Free PDF

Calculation

= 

= 

m = 4, n = 5 ⇒ m + n = 9 

Hence option 2 is correct

Calculation

P(2λ + 1, 3λ + 2, 4λ + 3) on L₁

Q(3µ + 2, 4µ + 4, 5µ + 5) on L₂

Dr’s of PQ = 3µ – 2λ + 1, 4µ – 3λ + 2, 5µ – 4λ + 2

PQ ⊥ L₁ 

⇒ (3µ – 2λ + 1)² + (4µ – 3λ + 2)3 + (5µ – 4λ + 2)4 = 0

38µ – 29λ + 16 = 0 …(1)

PQ ⊥ L₂ 

⇒ (3µ – 2λ + 1)3 + (4µ – 3λ + 2)4 + (5µ – 4λ + 2)5 = 0

50µ – 38λ + 21 = 0 …(2)

By (1) & (2) 

∴ 

Line PQ 

lies on the line PQ 

Hence option 4 is correct

Calculation

Let P be any point on the line,

P lies on the plane

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Line is

Another line is (line ) Shortest distance be d.

⇒  where are DRS of lines

⇒ 

Hence option 1 is correct

Concept Used:

The shortest distance between two skew lines is given by:

where and are points on the two lines, and and are the direction vectors of the two lines.

Calculation:

Given:

Skew lines:

Here, ,

and ,

Then

Now,

So,

And

Therefore,

Hence option 3 is correct

Calculation

Lines passed through the points

,

Shortest distance = 

Hence option 1 is correct

Concept: 

The magnitude of the shortest distance between the lines  and  is 

Given:  

The lines  and .

Rewriting the given equations,

 and 

⇒  ,   and  ,  

Therefore, the magnitude of the shortest distance between the given lines is

Therefore, the magnitude of the shortest distance between the given lines is .

Concept:

• If two lines are parallel, then the distance between them is fixed.
• The distance between two parallel lines  and  is given by the formula: .

Calculation:

Using the formula for the distance between two parallel lines, we can say that the distance is .

Concept:

The shortest distance between the skew line  is given by:

Calculation:

Given: Equation of lines is 

By comparing the given equations with , we get

⇒ x1 = 8, y1 = - 9, z1 = 10, a1 = 3, b1 = -16 and c1 = 7

Similarly, x2 = 15, y2 = 29, z2 = 5, a2 = 3, b2 = 8 and c2 = -5

So, 

As we know that shortest distance between two skew lines is given by:

⇒ SD = 14 units

Hence, option B is the correct answer.

Concept:

The shortest distance between parallel lines  and  is given by: 

Calculation:

L₁:  can be written as .

L₂:  can be written as .

Here, we see both lines are parallel and  ,  and .

 The shortest distance between parallel lines L₁ and L₂: 

⇒ 

⇒  ⇒  unit.

Hence, option 1 is correct.

Concept:

The shortest distance between parallel lines  and  is given by: 

Calculation:

L₁:  can be written as .

L₂:  can be written as .

Here, we see both lines are parallel and  ,  and .

 The shortest distance between parallel lines L₁ and L₂: 

⇒ 

⇒  ⇒  unit.

Hence, option 1 is correct.

Concept:

The shortest distance between the lines   and  is given by:

Calculation:

Here we have to find the shortest distance between the lines ​​ and 

Let line L₁ be represented by the equation  and line L₂ be represented by the equation 

⇒ x1 = 0, y1 = 2, z1 = 0  and a1 = -1, b1 = 0, c1 = 1.

⇒ x2 = -2, y2 = 0, z2 = 0  and a2 = 1, b2 = 1, c2 = 0.

∵ The shortest distance between the lines is given by:  

⇒     

⇒ 

⇒ d = 0

Hence, option 4 is correct.

Concept:

The shortest distance between parallel lines  and  is given by: 

Calculation:

Given: Equation of lines   and .

So, by comparing the above equations with  and  we get

⇒  ,   and .

 The shortest distance between parallel lines \(\vec{r}= \vec{a_{1}}+ \lambda \vec{b} \) and  is given by: 

⇒ 

⇒ 

⇒ 

⇒ k = 20

Hence, option 4 is correct.

Concept:

The shortest distance between the skew line  is given by:

Calculation:

Given: Equation of lines is 

By comparing the given equations with , we get

⇒ x₁ = - 3, y₁ = 6, z₁ = 0, a₁ = - 4, b₁ = 3 and c₁ = 2

Similarly, x₂ = - 2, y₂ = 0, z₂ = 7, a₂ = - 4, b₂ = 1 and c₂ = 1

So, 

Similarly, 

As we know that shortest distance between two skew lines is given by:

Concept -

Shortest distance between two lines is:

d = 

Explanation -

The given lines are :

 and 

So, 

, 

∴ 

= 

= 

Shortest distance, 

= 

=  units

Hence Option (2) is correct.

Concept:

The shortest distance between the lines   and  is given by:

Calculation:

Here we have to find the shortest distance between the lines ​​ and 

Let line L1 be represented by the equation  and line L2 be represented by the equation 

⇒ x1 = 5, y1 = -2, z1 = 0  and a1 = 7, b1 = -5, c1 = 1.

⇒ x2 = 0, y2 = 0, z2 = 0  and a2 = 1, b2 = 2, c2 = 3.

∵ The shortest distance between the lines is given by:  

⇒ 

⇒ 

⇒

⇒

Hence, option 3 is correct.