Class 12 RD Sharma Solutions – Chapter 28 The Straight Line in Space – Exercise 28.3

Last Updated : 21 Jul, 2021

Question 1. Show that the lines  and  intersect and find their point of intersection.

Solution:

Given that the coordinates of any point on the first line are

â‡’ x = Î», y = 2Î» + 2, z = 3Î» – 3

The coordinates of a general point on the second line are given by:

â‡’ x = 2Î¼ + 2, y = 3Î¼ + 6, z = 4Î¼ + 3

If the lines intersect, for some values of Î» and Î¼, we must have:

Î» – 2Î¼ = 2              ……(1)

2Î» – 3Î¼ = 4          ……(2)

3Î» – 4Î¼ = 6           …..(3)

Solving this system of equations, we get

Î» = 2 and Î¼ = 0

On substituting the values in eq(3), we have

LHS = 3(2) – 4(0)

= 6 = RHS

Thus, the given lines intersect at (2, 6, 3).

Question 2. Show that the lines  and  do not intersect.

Solution:

Given that the coordinates of any point on the first line are

â‡’ x = 3Î» + 1, y = 2Î» – 1, z = 5Î» + 1

The coordinates of a general point on the second line are given by:

â‡’ x = 4Î¼ – 2, y = 3Î¼ + 1, z = -2Î¼ – 1

If the lines intersect, for some values of Î» and Î¼, we must have:

3Î» – 4Î¼ = -3             ……(1)

2Î» – 3Î¼ = 2              ……(2)

5Î» + 2Î¼ = -2            …..(3)

Solving this system of equations, we get

Î» = -17 and Î¼ = -12

On substituting the values in eq(3), we have

LHS = 3(-17) + 2(-12)

= -75 â‰  RHS

Thus, the given lines do not intersect with each other.

Question 3. Show that the lines  and  intersect and find their point of intersection.

Solution:

Given that the coordinates of any point on the first line are

â‡’ x = 3Î» – 1, y = 5Î» – 3, z = 7Î» – 5

The coordinates of a general point on the second line are given by:

â‡’ x = 2Î¼ + 2, y = 3Î¼ + 6, z = 4Î¼ + 3

If the lines intersect, for some values of Î» and Î¼, we must have:

3Î» – Î¼ = 3               ……(1)

5Î» – 3Î¼ = 7            ……(2)

7Î» – 5Î¼ = 11           …..(3)

Solving this system of equations, we get

Î» = 1/2 and Î¼ = -3/2

On substituting the values in eq(3), we have

LHS = 3(2) – 4(0)

= -3/2 = RHS

Now put the value of Î» in first equation and we get

x = 1/2, y = -1/2, z = -3/2

Thus, the given lines intersect at (1/2, -1/2, -3/2).

Question 4. Prove that the line through (0, -1, -1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(-4, 4, 4). Also, find their point of intersection.

Solution:

Given that the coordinates of any point on the line AB are

â‡’ x = 4Î», y = 6Î» – 1, z = 2Î» – 1

Also, given that the coordinates of any point on the line CD are

â‡’ x = 7Î¼ + 3, y = 5Î¼ + 9, z = 4

If the lines intersect, for some values of Î» and Î¼, we must have:

4Î» – 7Î¼ = 3            ……(1)

6Î» – 5Î¼ = 10         ……(2)

Î» = 5/2                  …..(3)

â‡’ Î» = 5/2 and Î¼ = 1.

On substituting the values in eq(3), we have

LHS = 4(5/2) – 7(1)

= 3 = RHS

Now put the value of Î» in line AB, we get

x = 10, y = 14, z = 4

Thus, the given lines AB and CD intersect at point (10, 14, 4).

Question 5. Prove that the line  and  intersect and find their point of intersection.

Solution:

According to the question, it is given that the position vector of two points on the lines are

If the lines intersect, then for some value of Î» and Î¼, we must have:

Now equate the coefficient of we get

1 + 3Î» = 4 + 2Î¼    ……(1)

1 – Î» = 0                …..(2)

-1 = -1 +3Î¼           …..(3)

On solving the equation, we get

Î» = 1 and Î¼ = 0.

Now, substituting the values in eq(1), we get

1 + 3(1) = 4 + 2(0)

4 = 4

LHS = RHS

Thus, the coordinates of the point of intersection of the two lines are (4, 0, -1).

(i)  and

Solution:

Given that:

If the lines intersect, then for some value of Î» and Î¼, we must have:

Now equate the coefficient of we get

1 + 2Î» = 2 + Î¼      …..(1)

-1 = -1 + Î¼           …..(2)

Î» = -Î¼                 …..(3)

On solving the equations, we get

Î» = 0 and Î¼ = 0.

Now, substitute the values in eq(1), we get

1 + 2Î» = 2 + Î¼

1 + 2(0) = 2 + 0

1 â‰  2

LHS â‰  RHS

Thus, the given lines do not intersect.

(ii)  and

Solution:

Given that the coordinates of any point on the line AB are

â‡’ x = 2Î» + 1, y = 3Î» – 1, z = Î»

The coordinates of a general point on the second line are given by

â‡’ x = 5Î¼ – 1, y = Î¼ + 2, z = 2

If the lines intersect, for some values of Î» and Î¼, we must have:

2Î» – 5Î¼ = -2             ……(1)

3Î» – Î¼ = 3                ……(2)

Î» = 2                        …..(3)

Solving this system of equations, we get

Î» = 2 and Î¼ = 3

On substituting the values in eq(3), we have

LHS = 2(2) – 5(3)

= -2 â‰  RHS

Thus, the given lines do not intersect each other.

(iii)  and

Solution:

Given that the coordinates of any point on the line AB are

â‡’ x = Î», y = 2Î» + 2, z = 3Î» – 3

The coordinates of a general point on the second line are given by

â‡’ x = 2Î¼ + 4, y = 0, z = 3Î¼ – 1

If the lines intersect, for some values of Î» and Î¼, we must have:

Î» – 2Î¼ = 2            …….(1)

2Î» – 3Î¼ = 4         ……(2)

3Î» – 4Î¼ = 6         ……(3)

On solving this system of equations, we get

Î» = 1 and Î¼ = 0

On substituting the values in eq(3), we have

LHS = 3(1) – 2(0)

= 3 = RHS

Thus, the given lines intersect at (4, 0, -1).

(iv)  and

Solution:

Given that the coordinates of any point on the line AB are

â‡’ x = 4Î» + 5, y = 4Î» + 7, z = -5Î» – 3

The coordinates of a general point on the second line are given by:

â‡’ x = 7Î¼ + 8, y = Î¼ + 4, z = 3Î¼ + 5

If the lines intersect, for some values of Î» and Î¼, we must have:

4Î» – 7Î¼ = 3            …….(1)

4Î» – Î¼ = -3            ……(2)

5Î» + 3Î¼ = -8        ……(3)

On solving this system of equations, we get

Î» = -1 and Î¼ = -1

On substituting the values in eq(3), we have

LHS = 5(-1) – 3(-1)

= -8 = RHS

Thus, the given lines intersect at (1, 3, 2).

Question 7. Show that the lines  and  are intersecting. Hence, find their point of intersection.

Solution:

Given that,

If the lines intersect, then for some value of Î» and Î¼, we must have:

Now equate the coefficient of we get

3 + Î» = 5 + 3Î¼       ……..(1)

2 + 2Î» = -2 + 2Î¼   ……..(2)

2Î» – 4 = 6Î¼           ……..(3)

Solving the equation, we have:

Î» = -4 and Î¼ = -2.

On substituting the values, we get

LHS = 2(-4) – 4

= -12

RHS = 6(-2)

= -12

Thus, the given lines intersect at point(-1, -6, -12).

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