# RD Sharma solutions for Class 12 Maths chapter 29 - The Plane [Latest edition]

## Chapter 29: The Plane

Exercise 29.1Exercise 29.2Exercise 29.3Exercise 29.4Exercise 29.5Exercise 29.6Exercise 29.7Exercise 29.8Exercise 29.9Exercise 29.11Exercise 29.12Exercise 29.13Exercise 29.14Exercise 29.15Very Short AnswersMCQ
Exercise 29.1 [Pages 4 - 5]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.1 [Pages 4 - 5]

Exercise 29.1 | Q 1.1 | Page 4

Find the equation of the plane passing through the following points.

(2, 1, 0), (3, −2, −2) and (3, 1, 7)

Exercise 29.1 | Q 1.2 | Page 4

Find the equation of the plane passing through the following points.

(−5, 0, −6), (−3, 10, −9) and (−2, 6, −6)

Exercise 29.1 | Q 1.3 | Page 4

Find the equation of the plane passing through the following point

(1, 1, 1), (1, −1, 2) and (−2, −2, 2)

Exercise 29.1 | Q 1.4 | Page 4

Find the equation of the plane passing through the following points.

(2, 3, 4), (−3, 5, 1) and (4, −1, 2)

Exercise 29.1 | Q 1.5 | Page 4

Find the equation of the plane passing through the following point

(0, −1, 0), (3, 3, 0) and (1, 1, 1)

Exercise 29.1 | Q 2 | Page 5

Show that the four points (0, −1, −1), (4, 5, 1), (3, 9, 4) and (−4, 4, 4) are coplanar and find the equation of the common plane.

Exercise 29.1 | Q 3.1 | Page 5

Show that the following points are coplanar.
(0, −1, 0), (2, 1, −1), (1, 1, 1) and (3, 3, 0)

Exercise 29.1 | Q 3.2 | Page 5

Show that the following points are coplanar.

(0, 4, 3), (−1, −5, −3), (−2, −2, 1) and (1, 1, −1)

Exercise 29.1 | Q 4 | Page 5

Find the coordinates of the point where the line through (3, -4 , -5 ) and B (2, -3 , 1) crosses the plane passing through three points L(2,2,1), M(3,0,1) and N(4, -1,0 ) . Also, find the ratio in which diveides the line segment AB.

Exercise 29.2 [Page 7]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.2 [Page 7]

Exercise 29.2 | Q 1 | Page 7

Write the equation of the plane whose intercepts on the coordinate axes are 2, −3 and 4.

Exercise 29.2 | Q 2.1 | Page 7

Reduce the equations of the following planes to intercept form and find the intercepts on the coordinate axes.
4x + 3y − 6z − 12 = 0

Exercise 29.2 | Q 2.2 | Page 7

Reduce the equations of the following planes to intercept form and find the intercepts on the coordinate axes.

2x + 3y − z = 6

Exercise 29.2 | Q 2.3 | Page 7

Reduce the equations of the following planes to intercept form and find the intercepts on the coordinate axes.

2x − y + z = 5

Exercise 29.2 | Q 3 | Page 7

Find the equation of a plane which meets the axes at AB and C, given that the centroid of the triangle ABC is the point (α, β, γ).

Exercise 29.2 | Q 4 | Page 7

Find the equation of the plane passing through the point (2, 4, 6) and making equal intercepts on the coordinate axes.

Exercise 29.2 | Q 5 | Page 7

A plane meets the coordinate axes at AB and C, respectively, such that the centroid of triangle ABC is (1, −2, 3). Find the equation of the plane.

Exercise 29.3 [Pages 13 - 14]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.3 [Pages 13 - 14]

Exercise 29.3 | Q 1 | Page 13

Find the vector equation of a plane passing through a point with position vector $2 \hat{i} - \hat{j} + \hat{k}$ and perpendicular to the vector  $4 \hat{i} + 2 \hat{j} - 3 \hat{k} .$

Exercise 29.3 | Q 2.1 | Page 13

Find the Cartesian form of the equation of a plane whose vector equation is

$\vec{r} \cdot \left( 12 \hat{i} - 3 \hat{j} + 4 \hat{k} \right) + 5 = 0$

Exercise 29.3 | Q 2.2 | Page 13

Find the Cartesian form of the equation of a plane whose vector equation is

$\vec{r} \cdot \left( - \hat{i} + \hat{j} + 2 \hat{k} \right) = 9$

Exercise 29.3 | Q 3 | Page 13

Find the vector equations of the coordinate planes.

Exercise 29.3 | Q 4.1 | Page 13

Find the vector equation of each one of following planes.

2x − y + 2z = 8

Exercise 29.3 | Q 4.2 | Page 13

Find the vector equation of each one of following planes.

x + y − z = 5

Exercise 29.3 | Q 4.3 | Page 13

Find the vector equation of each one of following planes.

x + y = 3

Exercise 29.3 | Q 5 | Page 13

Find the vector and Cartesian equations of a plane passing through the point (1, −1, 1) and normal to the line joining the points (1, 2, 5) and (−1, 3, 1).

Exercise 29.3 | Q 6 | Page 13

$\vec{n}$ is a vector of magnitude $\sqrt{3}$ and is equally inclined to an acute angle with the coordinate axes. Find the vector and Cartesian forms of the equation of a plane which passes through (2, 1, −1) and is normal to $\vec{n}$ .

Exercise 29.3 | Q 7 | Page 13

The coordinates of the foot of the perpendicular drawn from the origin to a plane are (12, −4, 3). Find the equation of the plane.

Exercise 29.3 | Q 8 | Page 13

Find the equation of the plane passing through the point (2, 3, 1), given that the direction ratios of the normal to the plane are proportional to 5, 3, 2.

Exercise 29.3 | Q 9 | Page 13

If the axes are rectangular and P is the point (2, 3, −1), find the equation of the plane through P at right angles to OP.

Exercise 29.3 | Q 10 | Page 13

Find the intercepts made on the coordinate axes by the plane 2x + y − 2z = 3 and also find the direction cosines of the normal to the plane.

Exercise 29.3 | Q 11 | Page 13

A plane passes through the point (1, −2, 5) and is perpendicular to the line joining the origin to the point

$\text{ 3 } \hat{i} + \hat{j} - \hat{k} .$ Find the vector and Cartesian forms of the equation of the plane.

Exercise 29.3 | Q 12 | Page 13

Find the equation of the plane that bisects the line segment joining the points (1, 2, 3) and (3, 4, 5) and is at right angle to it.

Exercise 29.3 | Q 13.1 | Page 13

Show that the normals to the following pairs of planes are perpendicular to each other.

x − y + z − 2 = 0 and 3x + 2y − z + 4 = 0

Exercise 29.3 | Q 13.2 | Page 13

Show that the normals to the following pairs of planes are perpendicular to each other.

$\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + 3 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 2 \hat{i} - 2 \hat{j} - 2 \hat{k} \right) = 5$
Exercise 29.3 | Q 14 | Page 13

Show that the normal vector to the plane 2x + 2y + 2z = 3 is equally inclined to the coordinate axes.

Exercise 29.3 | Q 15 | Page 14

Find a vector of magnitude 26 units normal to the plane 12x − 3y + 4z = 1.

Exercise 29.3 | Q 16 | Page 14

If the line drawn from (4, −1, 2) meets a plane at right angles at the point (−10, 5, 4), find the equation of the plane.

Exercise 29.3 | Q 17 | Page 14

Find the equation of the plane which bisects the line segment joining the points (−1, 2, 3) and (3, −5, 6) at right angles.

Exercise 29.3 | Q 18 | Page 14

Find the vector and Cartesian equations of the plane that passes through the point (5, 2, −4) and is perpendicular to the line with direction ratios 2, 3, −1.

Exercise 29.3 | Q 19 | Page 14

If O be the origin and the coordinates of P be (1, 2,−3), then find the equation of the plane passing through P and perpendicular to OP.

Exercise 29.3 | Q 20 | Page 14

Find the vector equation of the plane with intercepts 3, –4 and 2 on xy and z-axis respectively.

Exercise 29.3 | Q 21 | Page 14

Find the vector equation of the plane with intercepts 3, –4 and 2 on xy and z-axis respectively.

Exercise 29.4 [Page 19]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.4 [Page 19]

Exercise 29.4 | Q 1 | Page 19

Find the vector equation of a plane which is at a distance of 3 units from the origin and has $\hat{k}$ as the unit vector normal to it.

Exercise 29.4 | Q 2 | Page 19

Find the vector equation of a plane which is at a distance of 5 units from the origin and which is normal to the vector  $\hat{i} - \text{2 } \hat{j} - \text{2 } \hat{k} .$

Exercise 29.4 | Q 3 | Page 19

Reduce the equation 2x − 3y − 6z = 14 to the normal form and, hence, find the length of the perpendicular from the origin to the plane. Also, find the direction cosines of the normal to the plane.

Exercise 29.4 | Q 4 | Page 19

Reduce the equation $\vec{r} \cdot \left( \hat{i} - 2 \hat{j} + 2 \hat{k} \right) + 6 = 0$ to normal form and, hence, find the length of the perpendicular from the origin to the plane.

Exercise 29.4 | Q 5 | Page 19

Write the normal form of the equation of the plane 2x − 3y + 6z + 14 = 0.

Exercise 29.4 | Q 6 | Page 19

The direction ratios of the perpendicular from the origin to a plane are 12, −3, 4 and the length of the perpendicular is 5. Find the equation of the plane.

Exercise 29.4 | Q 7 | Page 19

Find a unit normal vector to the plane x + 2y + 3z − 6 = 0.

Exercise 29.4 | Q 8 | Page 19

Find the equation of a plane which is at a distance of $3\sqrt{3}$  units from the origin and the normal to which is equally inclined to the coordinate axes.

Exercise 29.4 | Q 9 | Page 19

find the equation of the plane passing through the point (1, 2, 1) and perpendicular to the line joining the points (1, 4, 2) and (2, 3, 5). Find also the perpendicular distance of the origin from this plane

Exercise 29.4 | Q 10 | Page 19

Find the vector equation of the plane which is at a distance of $\frac{6}{\sqrt{29}}$ from the origin and its normal vector from the origin is  $2 \hat{i} - 3 \hat{j} + 4 \hat{k} .$ Also, find its Cartesian form.

Exercise 29.4 | Q 11 | Page 19

Find the distance of the plane 2x − 3y + 4z − 6 = 0 from the origin.

Exercise 29.5 [Pages 22 - 23]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.5 [Pages 22 - 23]

Exercise 29.5 | Q 1 | Page 22

Find the vector equation of the plane passing through the points (1, 1, 1), (1, −1, 1) and (−7, −3, −5).

Exercise 29.5 | Q 2 | Page 23

Find the vector equation of the plane passing through the points P (2, 5, −3), Q (−2, −3, 5) and R (5, 3, −3).

Exercise 29.5 | Q 3 | Page 23

Find the vector equation of the plane passing through points A (a, 0, 0), B (0, b, 0) and C(0, 0, c). Reduce it to normal form. If plane ABC is at a distance p from the origin, prove that $\frac{1}{p^2} = \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} .$

Exercise 29.5 | Q 4 | Page 23

Find the vector equation of the plane passing through the points (1, 1, −1), (6, 4, −5) and (−4, −2, 3).

Exercise 29.5 | Q 5 | Page 23

Find the vector equation of the plane passing through the points $3 \hat{i} + 4 \hat{j} + 2 \hat{k} , 2 \hat{i} - 2 \hat{j} - \hat{k} \text{ and } 7 \hat{i} + 6 \hat{k} .$

Exercise 29.6 [Page 29]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.6 [Page 29]

Exercise 29.6 | Q 1.1 | Page 29

Find the angle between the given planes. $\vec{r} \cdot \left( 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right) = 1 \text{ and } \vec{r} \cdot \left( - \hat{i} + \hat{j} \right) = 4$

Exercise 29.6 | Q 1.2 | Page 29

Find the angle between the given planes. $\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) = 6 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + 6 \hat{j} - 2 \hat{k} \right) = 9$

Exercise 29.6 | Q 1.3 | Page 29
Find the angle between the given planes.
$\vec{r} \cdot \left( 2 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( \hat{i} - 2 \hat{j} + 2 \hat{k} \right) = 9$

Exercise 29.6 | Q 2.1 | Page 29

Find the angle between the planes.

2x − y + z = 4 and x + y + 2z = 3

Exercise 29.6 | Q 2.2 | Page 29

Find the angle between the planes.

x + y − 2z = 3 and 2x − 2y + z = 5

Exercise 29.6 | Q 2.3 | Page 29

Find the angle between the planes.

x − y + z = 5 and x + 2y + z = 9

Exercise 29.6 | Q 2.4 | Page 29

Find the angle between the planes.
2x − 3y + 4z = 1 and − x + y = 4

Exercise 29.6 | Q 2.5 | Page 29

Find the angle between the planes.

2x + y − 2z = 5 and 3x − 6y − 2z = 7

Exercise 29.6 | Q 3.1 | Page 29

Show that the following planes are at right angles.

$\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( - \hat{i} - \hat{j} + \hat{k} \right) = 3$

Exercise 29.6 | Q 3.2 | Page 29

Show that the following planes are at right angles.

x − 2y + 4z = 10 and 18x + 17y + 4z = 49

Exercise 29.6 | Q 4.1 | Page 29

Determine the value of λ for which the following planes are perpendicular to each other.

$\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) = 7 \text{ and } \vec{r} \cdot \left( \lambda \hat{i} + 2 \hat{j} - 7 \hat{k} \right) = 26$

Exercise 29.6 | Q 4.2 | Page 29

Determine the value of λ for which the following planes are perpendicular to each ot

2x − 4y + 3z = 5 and x + 2y + λz = 5

Exercise 29.6 | Q 4.3 | Page 29

Determine the value of λ for which the following planes are perpendicular to each other.

3x − 6y − 2z = 7 and 2x + y − λz = 5

Exercise 29.6 | Q 5 | Page 29

Find the equation of a plane passing through the point (−1, −1, 2) and perpendicular to the planes 3x + 2y − 3z = 1 and 5x − 4y + z = 5.

Exercise 29.6 | Q 6 | Page 29

Obtain the equation of the plane passing through the point (1, −3, −2) and perpendicular to the planes x + 2y + 2z = 5 and 3x + 3y + 2z = 8.

Exercise 29.6 | Q 7 | Page 29

Find the equation of the plane passing through the origin and perpendicular to each of the planes x + 2y − z = 1 and 3x − 4y + z = 5.

Exercise 29.6 | Q 8 | Page 29

Find the equation of the plane passing through the points (1, −1, 2) and (2, −2, 2) and which is perpendicular to the plane 6x − 2y + 2z = 9.

Exercise 29.6 | Q 9 | Page 29

Find the equation of the plane passing through the points (2, 2, 1) and (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 1.

Exercise 29.6 | Q 10 | Page 29

Find the equation of the plane passing through the points whose coordinates are (−1, 1, 1) and (1, −1, 1) and perpendicular to the plane x + 2y + 2z = 5.

Exercise 29.6 | Q 11 | Page 29

Find the equation of the plane with intercept 3 on the y-axis and parallel to the ZOX plane.

Exercise 29.6 | Q 12 | Page 29

Find the equation of the plane that contains the point (1, −1, 2) and is perpendicular to each of the planes 2x + 3y − 2z = 5 and x + 2y − 3z = 8.

Exercise 29.6 | Q 13 | Page 29

Find the equation of the plane passing through (abc) and parallel to the plane  $\vec{r} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 2 .$

Exercise 29.6 | Q 14 | Page 29

Find the equation of the plane passing through the point (−1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.

Exercise 29.6 | Q 15 | Page 29

Find the vector equation of the plane through the points (2, 1, −1) and (−1, 3, 4) and perpendicular to the plane x − 2y + 4z = 10

Exercise 29.7 [Page 33]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.7 [Page 33]

Exercise 29.7 | Q 1.1 | Page 33

Find the vector equations of the following planes in scalar product form  $\left( \vec{r} \cdot \vec{n} = d \right):$ $\vec{r} = \left( 2 \hat{i} - \hat{k} \right) + \lambda \hat{i} + \mu\left( \hat{i} - 2 \hat{j} - \hat{k} \right)$

Exercise 29.7 | Q 1.2 | Page 33

Find the vector equations of the following planes in scalar product form  $\left( \vec{r} \cdot \vec{n} = d \right):$ $\vec{r} = \left( 1 + s - t \right) \hat{t} + \left( 2 - s \right) \hat{j} + \left( 3 - 2s + 2t \right) \hat{k}$

Exercise 29.7 | Q 1.3 | Page 33

Find the vector equations of the following planes in scalar product form  $\left( \vec{r} \cdot \vec{n} = d \right):$$\vec{r} = \left( \hat{i} + \hat{j} \right) + \lambda\left( \hat{i} + 2 \hat{j} - \hat{k} \right) + \mu\left( - \hat{i} + \hat{j} - 2 \hat{k} \right)$

Exercise 29.7 | Q 1.4 | Page 33

Find the vector equations of the following planes in scalar product form  $\left( \vec{r} \cdot \vec{n} = d \right):$$\vec{r} = \hat{i} - \hat{j} + \lambda\left( \hat{i} + \hat{j} + \hat{k} \right) + \mu\left( 4 \hat{i} - 2 \hat{j} + 3 \hat{k} \right)$

Exercise 29.7 | Q 2.1 | Page 33

Find the Cartesian forms of the equations of the following planes. $\vec{r} = \left( \hat{i} - \hat{j} \right) + s\left( - \hat{i} + \hat{j} + 2 \hat{k} \right) + t\left( \hat{i} + 2 \hat{j} + \hat{k} \right)$

Exercise 29.7 | Q 2.2 | Page 33

Find the Cartesian forms of the equations of the following planes.

$\vec{r} = \left( 1 + s + t \right) \hat{i} + \left( 2 - s + t \right) \hat{i} + \left( 3 - 2s + 2t \right) \hat{k}$

Exercise 29.7 | Q 3.1 | Page 33

Find the vector equation of the following planes in non-parametric form. $\vec{r} = \left( \lambda - 2\mu \right) \hat{i} + \left( 3 - \mu \right) \hat{j} + \left( 2\lambda + \mu \right) \hat{k}$

Exercise 29.7 | Q 3.2 | Page 33

Find the vector equation of the following planes in non-parametric form. $\vec{r} = \left( 2 \hat{i} + 2 \hat{j} - \hat{k} \right) + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) + \mu\left( 5 \hat{i} - 2 \hat{j} + 7 \hat{k} \right)$

Exercise 29.8 [Pages 39 - 40]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.8 [Pages 39 - 40]

Exercise 29.8 | Q 1 | Page 39

Find the equation of the plane which is parallel to 2x − 3y + z = 0 and which passes through (1, −1, 2).

Exercise 29.8 | Q 2 | Page 39

Find the equation of the plane through (3, 4, −1) which is parallel to the plane $\vec{r} \cdot \left( 2 \hat{i} - 3 \hat{j} + 5 \hat{k} \right) + 2 = 0 .$

Exercise 29.8 | Q 3 | Page 39

Find the equation of the plane passing through the line of intersection of the planes 2x − 7y + 4z − 3 = 0, 3x − 5y + 4z + 11 = 0 and the point (−2, 1, 3).

Exercise 29.8 | Q 4 | Page 39

Find the equation of the plane through the point $2 \hat{i} + \hat{j} - \hat{k}$ and passing through the line of intersection of the planes $\vec{r} \cdot \left( \hat{i} + 3 \hat{j} - \hat{k} \right) = 0 \text{ and } \vec{r} \cdot \left( \hat{j} + 2 \hat{k} \right) = 0 .$

Exercise 29.8 | Q 5 | Page 39

Find the equation of the plane passing through the line of intersection of the planes 2x − y = 0 and 3z − y = 0 and perpendicular to the plane 4x + 5y − 3z = 8

Exercise 29.8 | Q 6 | Page 39

Find the equation of the plane which contains the line of intersection of the planes x + 2y + 3z − 4 = 0 and 2x + y − z + 5 = 0 and which is perpendicular to the plane 5x + 3y − 6z+ 8 = 0.

Exercise 29.8 | Q 7 | Page 39

Find the equation of the plane through the line of intersection of the planes x + 2y + 3z + 4 = 0 and x − y + z + 3 = 0 and passing through the origin.

Exercise 29.8 | Q 8 | Page 39

Find the vector equation (in scalar product form) of the plane containing the line of intersection of the planes x − 3y + 2z − 5 = 0 and 2x − y + 3z − 1 = 0 and passing through (1, −2, 3).

Exercise 29.8 | Q 9 | Page 39

Find the equation of the plane that is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z − 4 = 0, 2x + y − z + 5 = 0.

Exercise 29.8 | Q 10 | Page 39

Find the equation of the plane through the line of intersection of the planes  $\vec{r} \cdot \left( \hat{i} + 3 \hat{j} \right) + 6 = 0 \text{ and } \vec{r} \cdot \left( 3 \hat{i} - \hat{j} - 4 \hat{k} \right) = 0,$ which is at a unit distance from the origin.

Exercise 29.8 | Q 11 | Page 39

Find the equation of the plane passing through the intersection of the planes 2x + 3y − z+ 1 = 0 and x + y − 2z + 3 = 0 and perpendicular to the plane 3x − y − 2z − 4 = 0.

Exercise 29.8 | Q 12 | Page 39

Find the equation of the plane that contains the line of intersection of the planes  $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) - 4 = 0 \text{ and } \vec{r} \cdot \left( 2 \hat{i} + \hat{j} - \hat{k} \right) + 5 = 0$ and which is perpendicular  to the plane $\vec{r} \cdot \left( 5 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) + 8 = 0 .$

Exercise 29.8 | Q 13 | Page 39

Find the equation of the plane passing through (abc) and parallel to the plane $\vec{r} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 2 .$

Exercise 29.8 | Q 14 | Page 39

Find the equation of the plane passing through the intersection of the planes  $\vec{r} \cdot \left( 2 \hat{i} + \hat{j} + 3 \hat{k} \right) = 7, \vec{r} \cdot \left( 2 \hat{i} + 5 \hat{j} + 3 \hat{k} \right) = 9$ and the point (2, 1, 3).

Exercise 29.8 | Q 15 | Page 40

Find the equation of the plane through the intersection of the planes 3x − y + 2z = 4 and x + y + z = 2 and the point (2, 2, 1).

Exercise 29.8 | Q 16 | Page 40

Find the vector equation of the plane through the line of intersection of the planes x + yz = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x − y + z = 0.

Exercise 29.8 | Q 17 | Page 40

Find the vector equation of the plane passing through the intersection of the planes

$\vec{r} \cdot \left( \hat{ i } + \hat{ j }+ \hat{ k }\right) = \text{ 6 and }\vec{r} \cdot \left( \text{ 2 } \hat{ i} +\text{ 3 } \hat{ j } + \text{ 4 } \hat{ k } \right) = - 5$ and the point (1, 1, 1).

Exercise 29.8 | Q 18 | Page 40

Find the equation of the plane which contains the line of intersection of the planes x $+$  2y $+$  3 $z -$  4 $=$  0 and 2 $x + y - z$ $+$ 5  $=$ 0 and whose x-intercept is twice its z-intercept. Hence, write the equation of the plane passing through the point (2, 3,  $-$ 1) and parallel to the plane obtained above.

Exercise 29.8 | Q 19 | Page 40

Find the equation of the plane through the line of intersection of the planes $x + y + z =$1 and 2x $+$ 3 $+$ y $+$ 4$z =$ 5 and twice of its $y$ -intercept is equal to three times its $z$-intercept

Exercise 29.9 [Page 49]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.9 [Page 49]

Exercise 29.9 | Q 1 | Page 49

Find the distance of the point  $2 \hat{i} - \hat{j} - 4 \hat{k}$  from the plane  $\vec{r} \cdot \left( 3 \hat{i} - 4 \hat{j} + 12 \hat{k} \right) - 9 = 0 .$

Exercise 29.9 | Q 2 | Page 49

Show that the points $\hat{i} - \hat{j} + 3 \hat{k} \text{ and } 3 \hat{i} + 3 \hat{j} + 3 \hat{k}$ are equidistant from the plane $\vec{r} \cdot \left( 5 \hat{i} + 2 \hat{j} - 7 \hat{k} \right) + 9 = 0 .$

Exercise 29.9 | Q 3 | Page 49

Find the distance of the point (2, 3, −5) from the plane x + 2y − 2z − 9 = 0.

Exercise 29.9 | Q 4 | Page 49

Find the equations of the planes parallel to the plane x + 2y − 2z + 8 = 0 that are at a distance of 2 units from the point (2, 1, 1).

Exercise 29.9 | Q 5 | Page 49

Show that the points (1, 1, 1) and (−3, 0, 1) are equidistant from the plane 3x + 4y − 12z + 13 = 0.

Exercise 29.9 | Q 6 | Page 49

Find the equations of the planes parallel to the plane x − 2y + 2z − 3 = 0 and which are at a unit distance from the point (1, 1, 1).

Exercise 29.9 | Q 7 | Page 49

Find the distance of the point (2, 3, 5) from the xy - plane.

Exercise 29.9 | Q 8 | Page 49

Find the distance of the point (3, 3, 3) from the plane $\vec{r} \cdot \left( 5 \hat{i} + 2 \hat{j} - 7k \right) + 9 = 0$

Exercise 29.9 | Q 9 | Page 49

If the product of the distances of the point (1, 1, 1) from the origin and the plane x − y + z+ λ = 0 be 5, find the value of λ.

Exercise 29.9 | Q 10 | Page 49

Find an equation for the set of all points that are equidistant from the planes 3x − 4y + 12z = 6 and 4x + 3z = 7.

Exercise 29.9 | Q 11 | Page 49

Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, −3), B(−2, −3, 5) and C (5, 3, −3).

Exercise 29.9 | Q 12 | Page 49

A plane makes intercepts −6, 3, 4 respectively on the coordinate axes. Find the length of the perpendicular from the origin on it.

Exercise 29.9 | Q 13 | Page 49

Find the distance of the point (1, -2, 4) from plane passing throuhg the point (1, 2, 2) and perpendicular of the planes x - y + 2z = 3 and 2x - 2y + z + 12 = 0

Exercise 29.1 [Page 51]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.1 [Page 51]

Exercise 29.1 | Q 1 | Page 51

Find the distance between the parallel planes 2x − y + 3z − 4 = 0 and 6x − 3y + 9z + 13 = 0.

Exercise 29.1 | Q 2 | Page 51

Find the equation of the plane which passes through the point (3, 4, −1) and is parallel to the plane 2x − 3y + 5z + 7 = 0. Also, find the distance between the two planes.

Exercise 29.1 | Q 3 | Page 51

Find the equation of the plane mid-parallel to the planes 2x − 2y + z + 3 = 0 and 2x − 2y + z + 9 = 0.

Exercise 29.1 | Q 4 | Page 51

Find the distance between the planes $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) + 7 = 0 \text{ and } \vec{r} \cdot \left( 2 \hat{i} + 4 \hat{j} + 6 \hat{k} \right) + 7 = 0 .$

Exercise 29.11 [Pages 61 - 62]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.11 [Pages 61 - 62]

Exercise 29.11 | Q 1 | Page 61

Find the angle between the line $\vec{r} = \left( 2 \hat{i}+ 3 \hat {j} + 9 \hat{k} \right) + \lambda\left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right)$  and the plane  $\vec{r} \cdot \left( \hat{i} + \hat{j} + \hat{k} \right) = 5 .$

Exercise 29.11 | Q 2 | Page 61

Find the angle between the line $\frac{x - 1}{1} = \frac{y - 2}{- 1} = \frac{z + 1}{1}$  and the plane 2x + y − z = 4.

Exercise 29.11 | Q 3 | Page 61

Find the angle between the line joining the points (3, −4, −2) and (12, 2, 0) and the plane 3x − y + z = 1.

Exercise 29.11 | Q 4 | Page 61

The line  $\vec{r} = \hat{i} + \lambda\left( 2 \hat{i} - m \hat{j} - 3 \hat{k} \right)$  is parallel to the plane  $\vec{r} \cdot \left( m \hat{i} + 3 \hat{j} + \hat{k} \right) = 4 .$ Find m

Exercise 29.11 | Q 5 | Page 61

Show that the line whose vector equation is $\vec{r} = 2 \hat{i} + 5 \hat{j} + 7 \hat{k}+ \lambda\left( \hat{i} + 3 \hat{j} + 4 \hat{k} \right)$ is parallel to the plane whose vector  $\vec{r} \cdot \left( \hat{i} + \hat{j} - \hat{k} \right) = 7 .$  Also, find the distance between them.

Exercise 29.11 | Q 6 | Page 61

Find the vector equation of the line through the origin which is perpendicular to the plane  $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) = 3 .$

Exercise 29.11 | Q 7 | Page 61

Find the equation of the plane through (2, 3, −4) and (1, −1, 3) and parallel to x-axis.

Exercise 29.11 | Q 8 | Page 61

Find the equation of a plane passing through the points (0, 0, 0) and (3, −1, 2) and parallel to the line $\frac{x - 4}{1} = \frac{y + 3}{- 4} = \frac{z + 1}{7} .$

Exercise 29.11 | Q 9 | Page 61

Find the vector and Cartesian equations of the line passing through (1, 2, 3) and parallel to the planes $\vec{r} \cdot \left( \hat{i} - \hat{j} + 2 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j} + 2 \hat{k} \right) = 6$

Exercise 29.11 | Q 10 | Page 61

Prove that the line of section of the planes 5x + 2y − 4z + 2 = 0 and 2x + 8y + 2z − 1 = 0 is parallel to the plane 4x − 2y − 5z − 2 = 0.

Exercise 29.11 | Q 11 | Page 61

Find the vector equation of the line passing through the point (1, −1, 2) and perpendicular to the plane 2x − y + 3z − 5 = 0.

Exercise 29.11 | Q 12 | Page 61

Find the equation of the plane through the points (2, 2, −1) and (3, 4, 2) and parallel to the line whose direction ratios are 7, 0, 6.

Exercise 29.11 | Q 13 | Page 61

Find the angle between the line $\frac{x - 2}{3} = \frac{y + 1}{- 1} = \frac{z - 3}{2}$ and the plane

3x + 4y + z + 5 = 0.

Exercise 29.11 | Q 14 | Page 61

Find the equation of the plane passing through the intersection of the planes x − 2y + z = 1 and 2x + y + z = 8 and parallel to the line with direction ratios proportional to 1, 2, 1. Also, find the perpendicular distance of (1, 1, 1) from this plane

Exercise 29.11 | Q 15 | Page 61

State when the line $\vec{r} = \vec{a} + \lambda \vec{b}$  is parallel to the plane  $\vec{r} \cdot \vec{n} = d .$Show that the line  $\vec{r} = \hat{i} + \hat{j} + \lambda\left( 3 \hat{i} - \hat{j} + 2 \hat{k} \right)$  is parallel to the plane  $\vec{r} \cdot \left( 2 \hat{j} + \hat{k} \right) = 3 .$   Also, find the distance between the line and the plane.

Exercise 29.11 | Q 16 | Page 61

Show that the plane whose vector equation is $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} - \hat{k} \right) = 1$ and the line whose vector equation is  $\vec{r} = \left( - \hat{i} + \hat{j} + \hat{k} \right) + \lambda\left( 2 \hat{i} + \hat{j} + 4 \hat{k} \right)$   are parallel. Also, find the distance between them.

Exercise 29.11 | Q 17 | Page 61

Find the equation of the plane through the intersection of the planes 3x − 4y + 5z = 10 and 2x + 2y − 3z = 4 and parallel to the line x = 2y = 3z.

Exercise 29.11 | Q 18 | Page 62

Find the vector and Cartesian forms of the equation of the plane passing through the point (1, 2, −4) and parallel to the lines $\vec{r} = \left( \hat{i} + 2 \hat{j} - 4 \hat{k} \right) + \lambda\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right)$ and $\vec{r} = \left( \hat{i} - 3 \hat{j} + 5 \hat{k} \right) + \mu\left( \hat{i} + \hat{j} - \hat{k} \right)$ Also, find the distance of the point (9, −8, −10) from the plane thus obtained.

Exercise 29.11 | Q 19 | Page 62

Find the equation of the plane passing through the points (3, 4, 1) and (0, 1, 0) and parallel to the line

$\frac{x + 3}{2} = \frac{y - 3}{7} = \frac{z - 2}{5} .$

Exercise 29.11 | Q 20 | Page 62

Find the coordinates of the point where the line  $\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{2}$   intersects the plane x − y + z − 5 = 0. Also, find the angle between the line and the plane.

Exercise 29.11 | Q 21 | Page 62

Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} - 5 \hat{k} \right) + 9 = 0 .$

Exercise 29.11 | Q 22 | Page 62

Find the angle between the line

$\frac{x + 1}{2} = \frac{y}{3} = \frac{z - 3}{6}$  and the plane 10x + 2y − 11z = 3.

Exercise 29.11 | Q 23 | Page 62

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes  $\vec{r} \cdot \left( \hat{i} - \hat{j} + 2 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j} + \hat{k} \right) = 6 .$

Exercise 29.11 | Q 24 | Page 62

Find the value of λ such that the line $\frac{x - 2}{6} = \frac{y - 1}{\lambda} = \frac{z + 5}{- 4}$  is perpendicular to the plane 3x − y − 2z = 7.

Exercise 29.11 | Q 25 | Page 62

Find the equation of the plane passing through the points (−1, 2, 0), (2, 2, −1) and parallel to the line $\frac{x - 1}{1} = \frac{2y + 1}{2} = \frac{z + 1}{- 1}$

Exercise 29.12 [Page 65]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.12 [Page 65]

Exercise 29.12 | Q 1.1 | Page 65

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the   yz - plane .

Exercise 29.12 | Q 1.2 | Page 65

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the  zx - plane .

Exercise 29.12 | Q 2 | Page 65

Find the coordinates of the point where the line through (3, −4, −5) and (2, −3, 1) crosses the plane 2x + y + z = 7.

Exercise 29.12 | Q 3 | Page 65

Find the distance of the point (−1, −5, −10) from the point of intersection of the line $\vec{r} = \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) + \lambda\left( 3 \hat{i}+ 4 \hat{j} + 2 \hat{k} \right)$ and the plane  $\vec{r} . \left( \hat{i} - \hat{j} + \hat{k} \right) = 5 .$

Exercise 29.12 | Q 4 | Page 65

Find the distance of the point (2, 12, 5) from the point of intersection of the line $\vec{r} = 2 \hat{i} - 4 \hat{j}+ 2 \hat{k} + \lambda\left( 3 \hat{i} + 4 \hat{j} + 2 \hat{k} \right)$ and $\vec{r} . \left( \hat{i} - 2 \hat{j} + \hat{k} \right) = 0$

Exercise 29.12 | Q 5 | Page 65

Find the distance of the point P(−1, −5, −10) from the point of intersection of the line joining the points A(2, −1, 2) and B(5, 3, 4) with the plane  $x - y + z = 5$ .

Exercise 29.12 | Q 6 | Page 65

Find the distance of the point P(3, 4, 4) from the point, where the line joining the points A(3, −4, −5) and B(2, −3, 1) intersects the plane 2x + y + z = 7.

Exercise 29.12 | Q 7 | Page 65

Find the distance of the point (1, -5, 9) from the plane

$x - y + z =$ 5  measured along the line $x = y = z$  .

Exercise 29.13 [Pages 73 - 74]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.13 [Pages 73 - 74]

Exercise 29.13 | Q 1 | Page 73

Show that the lines $\vec{r} = \left( 2 \hat{j} - 3 \hat{k} \right) + \lambda\left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) \text{ and } \vec{r} = \left( 2 \hat{i} + 6 \hat{j} + 3 \hat{k} \right) + \mu\left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right)$  are coplanar. Also, find the equation of the plane containing them.

Exercise 29.13 | Q 2 | Page 74

Show that the lines $\frac{x + 1}{- 3} = \frac{y - 3}{2} = \frac{z + 2}{1} \text{ and }\frac{x}{1} = \frac{y - 7}{- 3} = \frac{z + 7}{2}$  are coplanar. Also, find the equation of the plane containing them.

Exercise 29.13 | Q 3 | Page 74

Find the equation of the plane containing the line $\frac{x + 1}{- 3} = \frac{y - 3}{2} = \frac{z + 2}{1}$  and the point (0, 7, −7) and show that the line  $\frac{x}{1} = \frac{y - 7}{- 3} = \frac{z + 7}{2}$ also lies in the same plane.

Exercise 29.13 | Q 4 | Page 74

Find the equation of the plane which contains two parallel lines$\frac{x - 4}{1} = \frac{y - 3}{- 4} = \frac{z - 2}{5}\text{ and }\frac{x - 3}{1} = \frac{y + 2}{- 4} = \frac{z}{5} .$

Exercise 29.13 | Q 5 | Page 74

Show that the lines  $\frac{x + 4}{3} = \frac{y + 6}{5} = \frac{z - 1}{- 2}$ and 3x − 2y + z + 5 = 0 = 2x + 3y + 4z − 4 intersect. Find the equation of the plane in which they lie and also their point of intersection.

Exercise 29.13 | Q 6 | Page 74

Show that the plane whose vector equation is $\vec{r} \cdot \left( \hat{i} + 2 \hat{j} - \hat{k} \right) = 3$ contains the line whose vector equation is $\vec{r} = \hat{i} + \hat{j} + \lambda\left( 2 \hat{i} + \hat{j} + 4 \hat{k} \right) .$

Exercise 29.13 | Q 7 | Page 74

Find the equation of the plane determined by the intersection of the lines $\frac{x + 3}{3} = \frac{y}{- 2} = \frac{z - 7}{6} \text{ and }\frac{x + 6}{1} = \frac{y + 5}{- 3} = \frac{z - 1}{2}$

Exercise 29.13 | Q 8 | Page 74

Find the vector equation of the plane passing through the points (3, 4, 2) and (7, 0, 6) and perpendicular to the plane 2x − 5y − 15 = 0. Also, show that the plane thus obtained contains the line $\vec{r} = \hat{i} + 3 \hat{j} - 2 \hat{k} + \lambda\left( \hat{i} - \hat{j} + \hat{k} \right) .$

Exercise 29.13 | Q 9 | Page 74

If the lines  $\frac{x - 1}{- 3} = \frac{y - 2}{- 2k} = \frac{z - 3}{2} \text{ and }\frac{x - 1}{k} = \frac{y - 2}{1} = \frac{z - 3}{5}$ are perpendicular, find the value of and, hence, find the equation of the plane containing these lines.

Exercise 29.13 | Q 10 | Page 74

Find the coordinates of the point where the line $\frac{x - 2}{3} = \frac{y + 1}{4} = \frac{z - 2}{2}$ intersect the plane x − y + z − 5 = 0. Also, find the angle between the line and the plane.

Exercise 29.13 | Q 11 | Page 74

Find the vector equation of the plane passing through three points with position vectors  $\hat{i} + \hat{j} - 2 \hat{k} , 2 \hat{i} - \hat{j} + \hat{k} \text{ and } \hat{i} + 2 \hat{j} + \hat{k} .$  Also, find the coordinates of the point of intersection of this plane and the line  $\vec{r} = 3 \hat{i} - \hat{j} - \hat{k} + \lambda\left( 2 \hat{i} - 2 \hat{j} + \hat{k} \right) .$

Exercise 29.13 | Q 12 | Page 74

Show that the lines  $\frac{5 - x}{- 4} = \frac{y - 7}{4} = \frac{z + 3}{- 5}$ and  $\frac{x - 8}{7} = \frac{2y - 8}{2} = \frac{z - 5}{3}$ are coplanar.

Exercise 29.13 | Q 13 | Page 74

Find the equation of a plane which passes through the point (3, 2, 0) and contains the line  $\frac{x - 3}{1} = \frac{y - 6}{5} = \frac{z - 4}{4}$ .

Exercise 29.13 | Q 14 | Page 74

Show that the lines  $\frac{x + 3}{- 3} = \frac{y - 1}{1} = \frac{z - 5}{5}$ and  $\frac{x + 1}{- 1} = \frac{y - 2}{2} = \frac{z - 5}{5}$  are coplanar. Hence, find the equation of the plane containing these lines.

Exercise 29.13 | Q 15 | Page 74

If the line $\frac{x - 3}{2} = \frac{y + 2}{- 1} = \frac{z + 4}{3}$  lies in the plane  $lx + my - z =$   then find the value of  $l^2 + m^2$ .

Exercise 29.13 | Q 16 | Page 74

Find the values of  $\lambda$ for which the lines

$\frac{x - 1}{1} = \frac{y - 2}{2} = \frac{z + 3}{\lambda^2}$and  $\frac{x - 3}{1} = \frac{y - 2}{\lambda^2} = \frac{z - 1}{2}$  are coplanar .
Exercise 29.13 | Q 17 | Page 74

If the lines  $x =$  5 ,  $\frac{y}{3 - \alpha} = \frac{z}{- 2}$ and   $x = \alpha$ $\frac{y}{- 1} = \frac{z}{2 - \alpha}$ are coplanar, find the values of  $\alpha$.

Exercise 29.13 | Q 18 | Page 74

If the straight lines  $\frac{x - 1}{2} = \frac{y + 1}{k} = \frac{z}{2}$ and $\frac{x + 1}{2} = \frac{y + 1}{2} = \frac{z}{k}$ are coplanar, find the equations of the planes containing them.

Exercise 29.14 [Page 77]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.14 [Page 77]

Exercise 29.14 | Q 1 | Page 77

Find the shortest distance between the lines

$\frac{x - 2}{- 1} = \frac{y - 5}{2} = \frac{z - 0}{3} \text{ and } \frac{x - 0}{2} = \frac{y + 1}{- 1} = \frac{z - 1}{2} .$

Exercise 29.14 | Q 2 | Page 77

Find the shortest distance between the lines

$\frac{x + 1}{7} = \frac{y + 1}{- 6} = \frac{z + 1}{1} \text{ and } \frac{x - 3}{1} = \frac{y - 5}{- 2} = \frac{z - 7}{1} .$

Exercise 29.14 | Q 3 | Page 77

Find the shortest distance between the lines

$\frac{x - 1}{2} = \frac{y - 3}{4} = \frac{z + 2}{1}$ and
$3x - y - 2z + 4 = 0 = 2x + y + z + 1$

Exercise 29.15 [Pages 81 - 82]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Exercise 29.15 [Pages 81 - 82]

Exercise 29.15 | Q 1 | Page 81

Find the image of the point (0, 0, 0) in the plane 3x + 4y − 6z + 1 = 0.

Exercise 29.15 | Q 2 | Page 81

Find the reflection of the point (1, 2, −1) in the plane 3x − 5y + 4z = 5.

Exercise 29.15 | Q 3 | Page 81

Find the coordinates of the foot of the perpendicular drawn from the point (5, 4, 2) to the line $\frac{x + 1}{2} = \frac{y - 3}{3} = \frac{z - 1}{- 1} .$

Hence, or otherwise, deduce the length of the perpendicular.

Exercise 29.15 | Q 4 | Page 81

Find the image of the point with position vector $3 \hat{i} + \hat{j} + 2 \hat{k}$  in the plane  $\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + \hat{k} \right) = 4 .$  Also, find the position vectors of the foot of the perpendicular and the equation of the perpendicular line through $3 \hat{i} + \hat{j} + 2 \hat{k} .$

Exercise 29.15 | Q 5 | Page 81

Find the coordinates of the foot of the perpendicular from the point (1, 1, 2) to the plane 2x − 2y + 4z + 5 = 0. Also, find the length of the perpendicular.

Exercise 29.15 | Q 6 | Page 82

Find the distance of the point (1, −2, 3) from the plane x − y + z = 5 measured along a line parallel to  $\frac{x}{2} = \frac{y}{3} = \frac{z}{- 6} .$

Exercise 29.15 | Q 7 | Page 82

Find the coordinates of the foot of the perpendicular from the point (2, 3, 7) to the plane 3x − y − z = 7. Also, find the length of the perpendicular.

Exercise 29.15 | Q 8 | Page 82

Find the image of the point (1, 3, 4) in the plane 2x − y + z + 3 = 0.

Exercise 29.15 | Q 9 | Page 82

Find the distance of the point with position vector

$- \hat{i} - 5 \hat{j} - 10 \hat{k}$  from the point of intersection of the line $\vec{r} = \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) + \lambda\left( 3 \hat{i} + 4 \hat{j} + 12 \hat{k} \right)$  with the plane $\vec{r} \cdot \left( \hat{i} - \hat{j}+ \hat{k} \right) = 5 .$

Exercise 29.15 | Q 10 | Page 82

Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane $\vec{r} \cdot \left( \hat{i} - 2 \hat{j} + 4 \hat{k} \right) + 5 = 0 .$

Exercise 29.15 | Q 11 | Page 82

Find the coordinates of the foot of the perpendicular and the perpendicular distance of the  point P (3, 2, 1) from the plane 2x − y + z + 1 = 0. Also, find the image of the point in the plane.

Exercise 29.15 | Q 12 | Page 82

Find the direction cosines of the unit vector perpendicular to the plane  $\vec{r} \cdot \left( 6 \hat{i} - 3 \hat{j} - 2 \hat{k} \right) + 1 = 0$ passing through the origin.

Exercise 29.15 | Q 13 | Page 82

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x − 3y + 4z − 6 = 0.

Exercise 29.15 | Q 14 | Page 82

Find the length and the foot of perpendicular from the point $\left( 1, \frac{3}{2}, 2 \right)$  to the plane $2x - 2y + 4z + 5 = 0$ .

Exercise 29.15 | Q 15 | Page 82

Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector $2 \hat{i} + 3 \hat{j} + 4 \hat{k}$ to the plane  $\vec{r} . \left( 2 \hat{i} + \hat{j} + 3 \hat{k} \right) - 26 = 0$ Also find image of P in the plane.

Very Short Answers [Pages 83 - 84]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane Very Short Answers [Pages 83 - 84]

Very Short Answers | Q 1 | Page 83

Write the equation of the plane parallel to XOY- plane and passing through the point (2, −3, 5).

Very Short Answers | Q 2 | Page 83

Write the equation of the plane parallel to the YOZ- plane and passing through (−4, 1, 0).

Very Short Answers | Q 3 | Page 83

Write the equation of the plane passing through points (a, 0, 0), (0, b, 0) and (0, 0, c).

Very Short Answers | Q 4 | Page 83

Write the general equation of a plane parallel to X-axis.

Very Short Answers | Q 5 | Page 83

Write the value of k for which the planes x − 2y + kz = 4 and 2x + 5y − z = 9 are perpendicular.

Very Short Answers | Q 6 | Page 83

Write the intercepts made by the plane 2x − 3y + 4z = 12 on the coordinate axes.

Very Short Answers | Q 7 | Page 83

Write the ratio in which the plane 4x + 5y − 3z = 8 divides the line segment joining the points (−2, 1, 5) and (3, 3, 2).

Very Short Answers | Q 8 | Page 83

Write the distance between the parallel planes 2x − y + 3z = 4 and 2x − y + 3z = 18.

Very Short Answers | Q 9 | Page 83

Write the plane  $\vec{r} \cdot \left( 2 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) = 14$  in normal form.

Very Short Answers | Q 10 | Page 83

Write the distance of the plane  $\vec{r} \cdot \left( 2 \hat{i} - \hat{j} + 2 \hat{k} \right) = 12$ from the origin.

Very Short Answers | Q 11 | Page 83

Write the equation of the plane  $\vec{r} = \vec{a} + \lambda \vec{b} + \mu \vec{c}$   in scalar product form.

Very Short Answers | Q 12 | Page 83

Write a vector normal to the plane  $\vec{r} = l \vec{b} + m \vec{c} .$

Very Short Answers | Q 13 | Page 83

Write the equation of the plane passing through (2, −1, 1) and parallel to the plane 3x + 2y −z = 7.

Very Short Answers | Q 14 | Page 83

Write the equation of the plane containing the lines $\vec{r} = \vec{a} + \lambda \vec{b} \text{ and } \vec{r} = \vec{a} + \mu \vec{c} .$

Very Short Answers | Q 15 | Page 83

Write the position vector of the point where the line $\vec{r} = \vec{a} + \lambda \vec{b}$ meets the plane  $\vec{r} . \vec{n} = 0 .$

Very Short Answers | Q 16 | Page 83

Write the value of k for which the line $\frac{x - 1}{2} = \frac{y - 1}{3} = \frac{z - 1}{k}$  is perpendicular to the normal to the plane  $\vec{r} \cdot \left( 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \right) = 4 .$

Very Short Answers | Q 17 | Page 84

Write the angle between the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z + 3}{- 2}$  and the plane x + y + 4 = 0.

Very Short Answers | Q 18 | Page 84

Write the intercept cut off by the plane 2x + y − z = 5 on x-axis.

Very Short Answers | Q 19 | Page 84

Find the length of the perpendicular drawn from the origin to the plane 2x − 3y + 6z + 21 = 0.

Very Short Answers | Q 20 | Page 84

Write the vector equation of the line passing through the point (1, −2, −3) and normal to the plane $\vec{r} \cdot \left( 2 \hat{i} + \hat{j} + 2 \hat{k} \right) = 5 .$

Very Short Answers | Q 21 | Page 84

Find the vector equation of the plane, passing through the point (abc) and parallel to the plane $\vec{r} . \left( \hat{i} + \hat{j} + \hat{k} \right) = 2$

Very Short Answers | Q 22 | Page 84

Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is $2 \hat{i} - 3 \hat{j} + 6 \hat{k}$ .

Very Short Answers | Q 23 | Page 84

Write the equation of a plane which is at a distance of $5\sqrt{3}$ units from origin and the normal to which is equally inclined to coordinate axes.

MCQ [Pages 84 - 86]

### RD Sharma solutions for Class 12 Maths Chapter 29 The Plane MCQ [Pages 84 - 86]

MCQ | Q 1 | Page 84

The plane 2x − (1 + λ) y + 3λz = 0 passes through the intersection of the planes

• 2x − y = 0 and y − 3z = 0

• 2x + 3z = 0 and y = 0

• 2x − y + 3z = 0 and y − 3z = 0

• None of these

MCQ | Q 2 | Page 84

The acute angle between the planes 2x − y + z = 6 and x + y + 2z = 3 is

•  45°

• 60°

•  30°

•  75°

MCQ | Q 3 | Page 84

The equation of the plane through the intersection of the planes x + 2y + 3z = 4 and 2x + y − z = −5 and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is

• 7x − 2y + 3z + 81 = 0

• 23x + 14y − 9z + 48 = 0

•  51x − 15y − 50z + 173 = 0

•  None of these

MCQ | Q 4 | Page 84

The distance between the planes 2x + 2y − z + 2 = 0 and 4x + 4y − 2z + 5 = 0 is

• $\frac{1}{2}$

• $\frac{1}{4}$

•  $\frac{1}{6}$

• None of these

MCQ | Q 5 | Page 84

The image of the point (1, 3, 4) in the plane 2x − y + z + 3 = 0 is

•  (3, 5, 2)

•  (−3, 5, 2)

•  (3, 5, −2)

• (3, −5, 2)

MCQ | Q 6 | Page 85

The equation of the plane containing the two lines

$\frac{x - 1}{2} = \frac{y + 1}{- 1} = \frac{z - 0}{3} \text{ and }\frac{x}{- 2} = \frac{y - 2}{- 3} = \frac{z + 1}{- 1}$

•  8x + y − 5z − 7 = 0

•  8x + y + 5z − 7 = 0

• 8x − y − 5z − 7 = 0

•  None of these

MCQ | Q 7 | Page 85

The equation of the plane $\vec{r} = \hat{i} - \hat{j} + \lambda\left( \hat{i} + \hat{j} + \hat{k} \right) + \mu\left( \hat{i} - 2 \hat{j} + 3 \hat{k} \right)$  in scalar product form is

•   $\vec{r} \cdot \left( 5 \hat{i} - 2 \hat{j} - 3 \hat{k} \right) = 7$

•  $\vec{r} \cdot \left( 5 \hat{i} + 2 \hat{j} - 3 \hat{k} \right) = 7$

•  $\vec{r} \cdot \left( 5 \hat{i} - 2 \hat{j} + 3 \hat{k} \right) = 7$

•  None of these

MCQ | Q 8 | Page 85

The distance of the line $\vec{r} = 2 \hat{i} - 2 \hat{j} + 3 \hat{k} + \lambda\left( \hat{i} - \hat{j}+ 4 \hat{k} \right)$  from the plane $\vec{r} \cdot \left( \hat{i} + 5 \hat{j} + \hat{k} \right) = 5$ is

• $\frac{5}{3\sqrt{3}}$

• $\frac{10}{3\sqrt{3}}$

• $\frac{25}{3\sqrt{3}}$

•  None of these

MCQ | Q 9 | Page 85

The equation of the plane through the line x + y + z + 3 = 0 = 2x − y + 3z + 1 and parallel to the line $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ is

•  x − 5y + 3z = 7

• x − 5y + 3z = −7

•  x + 5y + 3z = 7

•  x + 5y + 3z = −7

MCQ | Q 10 | Page 85

The vector equation of the plane containing the line $\vec{r} = \left( - 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right) + \lambda\left( 3 \hat{i} - 2 \hat{j} - \hat{k} \right)$ and the point  $\hat{i} + 2 \hat{j} + 3 \hat{k}$  is

• $\vec{r} \cdot \left( \hat{i} + 3 \hat{k} \right) = 10$

•  $\vec{r} \cdot \left( \hat{i} - 3 \hat{k} \right) = 10$

•  $\vec{r} \cdot \left( 3\hat{i} - \hat{k} \right) = 10$

• None of these

MCQ | Q 11 | Page 85

A plane meets the coordinate axes at AB and C such that the centroid of ∆ABC is the point (abc). If the equation of the plane is $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = k,$ then k =

•  1

•  2

•  3

•  None of these

MCQ | Q 12 | Page 85
The distance between the point (3, 4, 5) and the point where the line $\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z - 5}{2}$ meets the plane x + y + z = 17 is
•  1

• 2

•  3

• None of these

MCQ | Q 13 | Page 85

A vector parallel to the line of intersection of the planes$\vec{r} \cdot \left( 3 \hat{i} - \hat{j} + \hat{k} \right) = 1 \text{ and } \vec{r} \cdot \left( \hat{i} + 4 \hat{j} - 2 \hat{k} \right) = 2$ is

•  $- 2 \hat{i} + 7 \hat{j}+ 13 \hat{k}$

•   $2 \hat{i} + 7 \hat{j} - 13 \hat{k}$

•  $-2 \hat{i} + 7 \hat{j} + 13 \hat{k}$

•  $2 \hat{i} + 7 \hat{j} + 13 \hat{k}$

MCQ | Q 14 | Page 85

If a plane passes through the point (1, 1, 1) and is perpendicular to the line $\frac{x - 1}{3} = \frac{y - 1}{0} = \frac{z - 1}{4}$ then its perpendicular distance from the origin is

• 3/4

•  4/3

•  7/5

•  1

MCQ | Q 15 | Page 85

The equation of the plane parallel to the lines x − 1 = 2y − 5 = 2z and 3x = 4y − 11 = 3z − 4 and passing through the point (2, 3, 3) is

•  x − 4y + 2z + 4 = 0

• x + 4y + 2z + 4 = 0

•  x − 4y + 2z − 4 = 0

• None of these

MCQ | Q 16 | Page 86

The distance of the point (−1, −5, −10) from the point of intersection of the line $\vec{r} = 2 \hat{i}- \hat{j} + 2 \hat{k} + \lambda\left( 3 \hat{i} + 4 \hat{j}+ 12 \hat{k} \right)$   and the plane $\vec{r} \cdot \left( \hat{i} - \hat{j} + \hat{k} \right) = 5$ is

• 9

•  13

•  17

•  None of these

MCQ | Q 17 | Page 86

The equation of the plane through the intersection of the planes ax + by + cz + d = 0 andlx + my + nz + p = 0 and parallel to the line y=0, z=0

• (bl − amy + (cl − anz + dl − ap = 0

•  (am − blx + (mc − bnz + md − bp = 0

•  (na − clx + (bn − cmy + nd − cp = 0

• None of these

MCQ | Q 18 | Page 86

The equation of the plane which cuts equal intercepts of unit length on the coordinate axes is

•  x + y + z = 1

•  x + y + z = 0

• x + y − z = 1

•  x + y + z = 2

## Chapter 29: The Plane

Exercise 29.1Exercise 29.2Exercise 29.3Exercise 29.4Exercise 29.5Exercise 29.6Exercise 29.7Exercise 29.8Exercise 29.9Exercise 29.11Exercise 29.12Exercise 29.13Exercise 29.14Exercise 29.15Very Short AnswersMCQ

## RD Sharma solutions for Class 12 Maths chapter 29 - The Plane

RD Sharma solutions for Class 12 Maths chapter 29 (The Plane) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Maths chapter 29 The Plane are Three - Dimensional Geometry Examples and Solutions, Introduction of Three Dimensional Geometry, Equation of a Plane Passing Through Three Non Collinear Points, Relation Between Direction Ratio and Direction Cosines, Intercept Form of the Equation of a Plane, Coplanarity of Two Lines, Distance of a Point from a Plane, Angle Between Line and a Plane, Angle Between Two Planes, Angle Between Two Lines, Vector and Cartesian Equation of a Plane, Equation of a Plane in Normal Form, Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point, Plane Passing Through the Intersection of Two Given Planes, Shortest Distance Between Two Lines, Equation of a Line in Space, Direction Cosines and Direction Ratios of a Line.

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