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

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 = 2ÎĽ + 2, y = 3ÎĽ + 6, z = 4ÎĽ + 3

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

Î» – 2ÎĽ = 2              ……(1)

2Î» – 3ÎĽ = 4          ……(2)

3Î» – 4ÎĽ = 6           …..(3)

Solving this system of equations, we get

Î» = 2 and ÎĽ = 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 = 4ÎĽ – 2, y = 3ÎĽ + 1, z = -2ÎĽ – 1

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

3Î» – 4ÎĽ = -3             ……(1)

2Î» – 3ÎĽ = 2              ……(2)

5Î» + 2ÎĽ = -2            …..(3)

Solving this system of equations, we get

Î» = -17 and ÎĽ = -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 = 2ÎĽ + 2, y = 3ÎĽ + 6, z = 4ÎĽ + 3

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

3Î» – ÎĽ = 3               ……(1)

5Î» – 3ÎĽ = 7            ……(2)

7Î» – 5ÎĽ = 11           …..(3)

Solving this system of equations, we get

Î» = 1/2 and ÎĽ = -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 = 7ÎĽ + 3, y = 5ÎĽ + 9, z = 4

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

4Î» – 7ÎĽ = 3            ……(1)

6Î» – 5ÎĽ = 10         ……(2)

Î» = 5/2                  …..(3)

â‡’ Î» = 5/2 and ÎĽ = 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 ÎĽ, we must have:

Now equate the coefficient of we get

1 + 3Î» = 4 + 2ÎĽ    ……(1)

1 – Î» = 0                …..(2)

-1 = -1 +3ÎĽ           …..(3)

On solving the equation, we get

Î» = 1 and ÎĽ = 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 ÎĽ, we must have:

Now equate the coefficient of we get

1 + 2Î» = 2 + ÎĽ      …..(1)

-1 = -1 + ÎĽ           …..(2)

Î» = -ÎĽ                 …..(3)

On solving the equations, we get

Î» = 0 and ÎĽ = 0.

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

1 + 2Î» = 2 + ÎĽ

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 = 5ÎĽ – 1, y = ÎĽ + 2, z = 2

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

2Î» – 5ÎĽ = -2             ……(1)

3Î» – ÎĽ = 3                ……(2)

Î» = 2                        …..(3)

Solving this system of equations, we get

Î» = 2 and ÎĽ = 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 = 2ÎĽ + 4, y = 0, z = 3ÎĽ – 1

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

Î» – 2ÎĽ = 2            …….(1)

2Î» – 3ÎĽ = 4         ……(2)

3Î» – 4ÎĽ = 6         ……(3)

On solving this system of equations, we get

Î» = 1 and ÎĽ = 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 = 7ÎĽ + 8, y = ÎĽ + 4, z = 3ÎĽ + 5

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

4Î» – 7ÎĽ = 3            …….(1)

4Î» – ÎĽ = -3            ……(2)

5Î» + 3ÎĽ = -8        ……(3)

On solving this system of equations, we get

Î» = -1 and ÎĽ = -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 ÎĽ, we must have:

Now equate the coefficient of we get

3 + Î» = 5 + 3ÎĽ       ……..(1)

2 + 2Î» = -2 + 2ÎĽ   ……..(2)

2Î» – 4 = 6ÎĽ           ……..(3)

Solving the equation, we have:

Î» = -4 and ÎĽ = -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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