Skew Lines MCQ Quiz - Objective Question with Answer for Skew Lines - Download Free PDF

Calculation: 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Now, 28α² = 28× 

Hence, the correct answer is 18. 

Calculation: 

 and 

Hence, the correct answer is Option 1.

Calculation: 

⇒ 

⇒ 

= 

= 

⇒ d = 

= 

Hence, the correct answer is Option 2.

Calculation:

Let the skew lines be

So that

⇒ d_1​ = (2, 3, 4), a_1​ = (p, 2, 1), d_2​ = (3, 4, 5) a_2​ = (−1, 0, 0)

Their shortest distance is

⇒ 

Now, 

⇒   = (-1, 2,-1) 

⇒ 

and  

⇒ 

Hence 

For the ellipse 

⇒ 

with (a

The latus rectum has length 

Hence, the correct answer is Option 3.

Calculation:

We have two Lines

⇒ 

⇒  

To find their shortest distance, form the determinant

⇒  

⇒ 

Foot of the perpendicular from P  to L₁

Take , A general point on L₁ ay be written as Q ( 

Then, 

Perpendicularity to L₁ direction d₁ = (0, 0, 1) gives 

⇒ 

Thus, 

⇒ 

Hence, the correct answer is Option 3.

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.