Class 12 RD Sharma Solutions – Chapter 29 The Plane – Exercise 29.13

Last Updated : 18 Feb, 2022

Question 1. Show thatandare coplanar. Also, find the equation of the plane containing them.

Solution:

We know, the lines  and  are coplanar if:

Since, , the lines are coplanar.

Equation of the plane containing them:

Question 2. Show that the linesandare coplanar. Also, find the equation of the plane containing them.

Solution:

We know the lines  and  are coplanar if,

So,

= 1(4 + 3) âˆ’ 4(âˆ’6 âˆ’ 1) âˆ’ 5(9 âˆ’ 2)

= 7 + 28 âˆ’ 35

= 0.

So the lines are coplanar.

Equation of the plane:

â‡’ 7x + 7y + 7z = 0.

Question 3. Find the equation of the plane containing the line and the point (0,7,-7) and show that the linealso lies in the same plane.

Solution:

We know the equation of a plane passing through a point (x1,y1,z1) is given by

a(xâˆ’x1) + b(yâˆ’y1) + c(zâˆ’z1) = 0            ……..(1)

Since the required plane passes through (0,7,-7), the equation becomes

ax + b(y âˆ’ 7) + c(z + 7) = 0                …….(2)

It also contains  and point is (âˆ’1,3,âˆ’2).

a(âˆ’1) + b(3 âˆ’ 7) + c(âˆ’2 + 7) = 0

â‡’ âˆ’a âˆ’ 4b + 5c = 0

Also, âˆ’3a + 2b + c = 0

Solving the equations, we get x + y + z = 0

So lies on the plane x + y + z = 0.

Question 4. Find the equation of the plane which contains two parallel lines and

Solution:

We know the equation of a plane passing through a point (x1,y1,z1) is given by

a(xâˆ’x1) + b(yâˆ’y1) + c(zâˆ’z1) = 0            ……..(1)

The required plane passes through (4,3,2). Hence,

a(x âˆ’ 4) + b(y âˆ’ 3) + c(z âˆ’ 2) = 0

It also passes through (3,-2,0). Hence,

a(3 âˆ’ 4) + b(âˆ’2 âˆ’ 3) + c(0 âˆ’ 2) = 0

â‡’ a + 5b + 2c = 0                …….(2)

Also, a âˆ’ 4b + 5c = 0             ……..(3)

Solving (2) and (3)  by cross multiplication, we get the equation of the plane as:

â‡’ 11x âˆ’ y âˆ’ 3z âˆ’ 35 = 0.

Question 5. Show that the lines and 3x âˆ’ 2y + 5 = 0 = 2x + 3y + 4z âˆ’ 4  intersect. Find the equation of the plane.

Solution:

Using a1a2 + b1b2 + c1c2 = 0, we get

3a âˆ’ 2b + c = 0                ….(1)

Also, 2a + 3b + 4c = 0.     ….(2)

Solving (1) and (2) by cross multiplication, we have

Hence, the equation of the plane is 45x âˆ’ 17y + 25z + 53 = 0

and the point of intersection is (2,4,âˆ’3).

Question 6. Show that the plane whose vector equation is  contains the line whose vector equation is

Solution:

Here,

= 2(1) +1(2) + 4(âˆ’1)

Now,

= 1(1) + 1(2) + 0(âˆ’1)

= 3

Hence, the given line lies on the plane.

Question 7. Find the equation of the plane determined by the intersection of the lines and

Solution:

Let the plane be ax + by + cz + d = 0

Since the plane passes through the intersection of the given lines, normal of the plane is perpendicular to the two lines.

â‡’ 3a âˆ’ 2b + 6c = 0

and, a âˆ’ 3b + 2c = 0

Using cross multiplication, we have

â‡’

Question 8. Find the vector equation of the plane passing through the points (3,4,2) and (7,0,6) and perpendicular to the plane 2x âˆ’ 5y âˆ’ 15 = 0. Also, show that the plane thus obtained contains the line

Solution:

Let the equation of the plane be

Since the plane passes through (3,4,2) and (7,0,6), we have

and

Since the required plane is perpendicular to 2x âˆ’ 5y âˆ’ 15 = 0, we have,

â‡’ b = 2.5a

Substituting the value of b in the above equations we have,

and

Solving the above equations, we have

a = 17/5, b = 17/2 and c = âˆ’17/3.

Substituting the values in the equation of plane, we obtain

5x + 2y âˆ’ 3z = 17.

Vector equation of the plane becomes: .

Question 9. If the lines andare perpendicular, find the values of k and also the equation the plane containing these lines.

Solution:

The direction ratios of the two lines are r1 = (âˆ’3,âˆ’2k,2) and r2 = (k,1,5).

Since the lines are perpendicular, we have

(âˆ’3,âˆ’2k,2).(k,1,5) = 0

â‡’ 3k + 2k âˆ’ 10 = 0

â‡’ 5k = 10

â‡’ k = 2

Now, equation of the plane containing the lines is:

â‡’ âˆ’22x + 19y + 5z + 31 = 0.

Question 10. Find the coordinates of the point where the line intersects the plane x âˆ’ y + z âˆ’ 5 = 0. Also, find the angle between the line and the plane.

Solution:

Any point on the given line is of the form (3k + 2, 4k âˆ’ 1, 2k + 2).

We have, (3k + 2) âˆ’ (4k âˆ’ 1) + (2k + 2) âˆ’ 5 = 0

â‡’ k = 0.

Thus, the coordinates of the point become (2,âˆ’1,2).

Let v be the angle between the line and the plane. Then,

Here, l = 3, m = 4, n =2, a =1, b = âˆ’1, c = 1.

Hence,

â‡’

â‡’

Question 11. Find the vector equation of the plane passing through three points with position vectors

Solution:

Let  A, B and C be the three given vectors respectively.

and,

Now,

â‡’

Equation of the plane is:

Coordinates of the points are (1,1,âˆ’2).

Question 12. Show that the lines andare coplanar.

Solution:

We know the lines  and  are coplanar if,

or,

= 3(12 + 5) + 3(12 + 35) + 8(4 âˆ’ 28)

= 0.

Hence the lines are coplanar.

Question 13. Find the equation of the plane which passes through the point (3,2,0) and contains the line

Solution:

Required equation of the plane passing through (3,2,0) is:

a(x âˆ’ 3) + b(y âˆ’ 2) + cz = 0            ……(1)

Since the plane also passes through the given line, we have

4b + 4c = 0               ……(2)

Also, the plane will be parallel so,

a + 5b + 4c = 0        ……(3)

Solving (2) and (3), we have

â‡’

â‡’ a = âˆ’z, b = z and c = âˆ’z

Putting the values in (1), we have

x âˆ’ y + z âˆ’ 1 = 0.

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