Class 12 RD Sharma Solutions - Chapter 29 The Plane - Exercise 29.14

Question 1. Find the shortest distance between the lines  and .

Solution: 

Let us consider

According to the equations line P₁ passes through the point P(2, 5, 0)

And the equation of a plane containing line P₂ is

a(x - 0) + b(y + 1) + c(z - 1) = 0          -(1)

Where 2a - b + 2c = 0

If it is parallel to line P₁ then

-a + 2b + 3c = 0

So, 

Now, substitute the value of a, b, c in the eq(1) we get

a(x - 0) + b(y + 1) + c(z - 1) = 0

-7(x - 0) - 8(y + 1) + 3(z - 1) = 0

-7x - 8y - 8 + 3z - 3 = 0

7x + 8y - 3z + 11 = 0          -(2)

So, this is the equation of the plane that contain line P₂ and parallel to line P₁.

Hence, the shortest distance between P₁ and P₂ = Distance between point P(2, 5, 0) and plane (2)

Question 2. Find the shortest distance between the lines  and .

Solution: 

Let us consider

Let us assume the equation of the plane containing P₁ is a(x + 1) + b(y + 1) + c(z+1) = 0

Plane is parallel to P₁ = 7a - 6b + c = 0          -(1)

Plane is parallel to P₂ = a - 2b + c = 0          -(2)

On solving eq(1) and eq(2), we get,

The equation of the plane is -4(x + 1) - 6(y + 1) - 8(z + 1) = 0

Final equation of plane is 4(x + 1) + 6(y + 1) + 8(z + 1) = 0 

Question 3. Find the shortest distance between the lines  and 3x - y - 2z + 4 = 0, 2x + y + z + 1 = 0.

Solution: 

The equation of a plane containing the line 3x - y - 2z + 4 = 0, 2x + y + z + 1 = 0 is 

x(2λ + 3) + y(λ - 1) + z(λ - 2) + λ + 4 = 0          -(1)

If it is parallel to the line  then,

2(2λ + 3) + 4(λ - 1) + (λ - 2) = 0

λ = 0

On putting λ = 0 in eq(1) we get,

3x - y - 2z + 4 = 0          -(2)

As this equation of the plane consist the second line and parallel to the first line.

It is clear that the line  passes through the point (1, 3, -2)

So, the shortest distance 'D' between the given lines is equal to the 

length of perpendicular from point (1, 3, -2) on the plane (2)

D = 

Summary

Exercise 29.14 in RD Sharma Class 12 typically deals with finding the equation of a plane under various conditions. Key concepts include:

• General equation of a plane: ax + by + cz + d = 0
• Planes passing through given points
• Planes parallel or perpendicular to given vectors
• Planes containing given lines
• Distance between a point and a plane
• Angle between two planes