Skew Lines MCQ Quiz in मराठी - Objective Question with Answer for Skew Lines - मोफत PDF डाउनलोड करा

Concept:

The shortest distance between two skew lines  and    is given by:

d = 

Explanation:

Given,   ---(1)

 and  ---(2)

The foot of the perpendicular from the point (4,3,8) on the line L₁ is (3,5,7).

So, point (3,5,7) satisfies the equation of the line L₁.

Substitute x=3, y=5, and z=7 in equation (1), we have

⇒ 

⇒ 

⇒

Now,  = 1

⇒ 3-a= l

⇒ a+l=3  ---(3)

And  =1

⇒ 7-b =4

⇒ b=3

Direction cosines of the line joining points (4,3,8) and (3,5,7) are (3-4,5-3,7-8) i.e.(-1,2, -1).

The direction cosines of the line L₁ are (l,3,4).

Both lines are perpendicular.

∴ l× (-1)+2× 3+(-1)× 4 =0

⇒ -l +6-4=0

⇒-l+2=0

⇒ l=2

Substitute the value of l in equation (3), we have

⇒ a+2=3

⇒ a =1

Substitute the values of a,b and l in equation (1),

and  

Direction cosines of the line L₁ and L₂ are (2,3,4) and (3,4,5) respectively.

Now, the shortest distance between line L₁ and L₂ is

d = 

=

=

= 

=

The shortest distance between lines L₁ and L₂ is .

Hence, the correct answer is option (1).

Concept:

Now, the shortest distance between two lines is given by 

Calculation:

Given:   = 24î + 36ĵ + 72k̂ with its magnitude as 84

 = 8î - 9ĵ + 10k̂ + λ (3î - 16ĵ + 7k̂)

⇒ = 8î - 9ĵ + 10k̂ and = (3î - 16ĵ + 7k̂) ...(i)

Also, ​ ​= 15î + 29ĵ + 5k̂ + μ(3î + 8ĵ - 5k)

 = 15î + 29ĵ + 5k̂ and  = (3î + 8ĵ - 5k̂) ....(ii)

Now, ( -  ) = (15 - 8)î + (29 + 9)ĵ + (5 - 10)k̂ 

= 7î + 38ĵ - 5k̂ 

∴ Shortest distance 

Hence, option 2 is the correct answer.

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 = -1, y1 = 1, z1 = 9, a1 = 2, b1 = 1 and c1 = - 3

Similarly, x2 = 3, y2 = -15, z2 = 9, a2 = 2, b2 = -7 and c2 = 5

So, 

And 

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

⇒ 

Hence, option D is the correct answer.

Calculation

Lines passed through the points

,

Shortest distance = 

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 = 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.

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 = 4, z1 = - 2, a1 = -1, b1 = 2 and c1 = 1

Similarly, x2 = 1, y2 = - 7, z2 = -2, a2 = 1, b2 = 3 and c2 = 2

So, 

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

⇒ 

Hence, option C is the correct answer.

Calculation:

Distance between skew lines  and  is given by:

d = 

Calculation:

Given, (x – λ)/1 = (y – 1/2)/(1/2) = z/(-1/2)

(x – λ)/2 = (y-1/2)/1 = z/(-1) …(1) Point on line = (λ, 1/2, 0)

x/1 = (y + 2λ)/1 = (z – λ)/1 …(2) Point on line = (0, -2λ, λ)

∴ Distance between skew lines = 

= 

= |-5λ – 3/2|/

= √7/(2√2) (Given)

⇒ |10λ + 3| = 7

⇒ 10λ + 3 = ± 7

⇒ λ = - 1 [∵ λ is an integer]

⇒ |λ| = 1

∴ The value of |λ| is 1. 

Concept:

The shortest distance between the lines  and  is given by d = 

Calculation:

Given,  and 

∴ a₁ =  and b₁ = 

a₂ =  and b₂ = 

⇒ a₂ – a₁ = 

∴ 

= 

⇒ 

∴  = 16[-4 + 16] = (16)(12)

⇒ d =  = 4√3

∴ The shortest distance is 4√3.

The correct answer is Option 2.

Calculation

⇒ 

⇒ 

⇒ 

⇒ 

Hence, Option (3) is correct

Calculation

Given

d₁ = shortest distance between L₁ & L₂ 

⇒ d₁= 

⇒ d₁ = 2

d₂ = shortest distance between L₃ & L₄

⇒  

Hence

 = 16