Решения Р. Д. Шармы класса 12 - Глава 29 Самолет - Упражнение 29.14

Вопрос 1. Найдите кратчайшее расстояние между линиями а также .

Решение:

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)

Вопрос 2. Найдите кратчайшее расстояние между линиями. а также .

Решение:

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 

Вопрос 3. Найдите кратчайшее расстояние между линиями. и 3x - y - 2z + 4 = 0, 2x + y + z + 1 = 0.

Решение:

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 =