Class 12 RD Sharma Mathematics Solutions - Chapter 29 The Plane - Exercise 29.9

Chapter 29 "The Plane" of RD Sharma's Class 12 mathematics textbook focuses on the study of planes in three-dimensional space. Exercise 29.9 specifically deals with problems related to the image of a point with respect to a plane. This exercise challenges students to apply their understanding of plane equations, distance formulas, and vector operations to solve more complex geometrical problems involving reflections and symmetry.

Important Formulas Related to Plane

General equation of a plane: Ax + By + Cz + D = 0

Distance formula between a point (x₀, y₀, z₀) and a plane Ax + By + Cz + D = 0:

d = |Ax₀ + By₀ + Cz₀ + D| / √(A² + B² + C²)

Vector form of a plane equation: r · n = p, where n is the normal vector and p is a constant

Image of a point P(x, y, z) in a plane ax + by + cz + d = 0 is given by:

P'(x', y', z') = P - 2((aP.x + bP.y + cP.z + d) / (a² + b² + c²))(a, b, c)

Class 12 RD Sharma Mathematics Solutions - Exercise 29.9

Question 1. Find the distance of the point  from the plane 

Solution:

As we know the distance of a point  from a plane  is given by:

Here,  and  is the plane.

Hence, 

⇒ 

= |-47/13| units 

= 47/13 units 

Hence, the distance of the point from the plane is 47/13 units.

Question 2. Show that the points  and  are equidistant from the plane 

Solution:

As we know the distance of a point  from a plane  is given by:

Let D₁ be the distance of  from the plane 

⇒ 

= 

= 9/√78 units                  .......(1)

Now, let D₂ be the distance between point  and the plane .

⇒ 

= 

= 9/√78 units                    ......(2)

From eq(1) and (2), we have

The given points are equidistant from the given plane.

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

Solution:

As we know the distance is given by:

⇒ 

= 

= 9/√9 

D = 3 units.

Question 4. Find the equations of the planes parallel to the plane x + 2y − 2z + 8 = 0 which are at a distance of 2 units from the point (2, 1, 1).

Solution:

The equation of a plane parallel to the given plane is x + 2y − 2z + p = 0.

As we know that the distance between a point and plane is given by:

Given, D = 2 units. Hence,

⇒ 

On squaring both sides, we have

⇒ 36 = (2 + p)²

⇒ 2 + p = 6           or         2 + p = −6

⇒ p = 4                 or           p = −8

Hence, the equations of the required planes are:

x + 2y − 2z + 4 = 0 and x + 2y − 2z − 8 = 0.

Question 5. Show that the points (1, 1, 1) and(−3, 0, 1) are equidistant from the plane 3x + 4y − 12z +13 = 0.

Solution:

We know that the distance between a point and plane is given by:

Let D₁ be the distance of the point (1,1,1) from the plane.

⇒ 

= 8/13 units 

Let D₂ be the distance of the point.

⇒ 

= 8/13 units 

Hence, the points are equidistant from the plane.

Question 6. Find the equation of the planes parallel to the plane x − 2y + 2z − 3 = 0 which are at a unit distance from the point (2, 1, 1).

Solution:

Equation of a plane parallel to the given plane is x − 2y + 2z + p = 0.

As we know that the distance between a point and plane is given by:

Given, D = 1 units. Hence,

⇒ 

On squaring both sides, we have

⇒ 9 = (1 + p)²

⇒ 1 + p = 3           or         1 + p = −3

⇒ p = 2                 or           p = −4

Hence, the equations of the required planes are:

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

Question 7. Find the distance of the point (2, 3, 5) from the xy-plane.

Solution:

As we know the distance of the point from the plane is given by:

D = 

= 

= 

D = 5 units

Question 8. Find the distance of the point (3, 3, 3) from the plane 

Solution:

As we know the distance of a point  from a plane  is given by:

D = 

= 

= 

D = 9/√78 units.

Question 9. If the product of distances of the point (1, 1, 1) from the origin and the plane x − y + z + p = 0 be 5, find p.

Solution:

The distance of the point (1, 1, 1) from the origin is 

Distance of (1, 1, 1) from the plane is 

Given: 

⇒ |1 + p| = 5

⇒ p = 4 or −6.

Question 10. Find an equation of the set of all points that are equidistant from the planes 3x − 4y + 12 = 6 and 4x + 3z = 7.

Solution:

= 

Now, 

= 

Given, D₁ = D₂

⇒ 

Hence, the equations become

37x₁ + 20y₁ − 21z₁ − 61 = 0 and 67x₁ + 20y₁ + 99z₁ − 121 = 0.

Question 11. Find the distance between the point (7, 2, 4) and the plane determined by the points A(−2, −3, 5) and C(5, 3, −3).

Solution:

The equations of the plane are given as:

−4a − 8b + 8c = 0 and 3a − 2b + 0c = 0

Solving the above set using cross multiplication method, we get

⇒ 

⇒ 

⇒ 

⇒ a = 2p, b = 3p, c = 4p

Thus, the equation of the plane becomes 2x + 3y + 4z − 7 = 0.

and, distance = √29 units.

Question 12. A plane makes intercepts −6, 3, 4 respectively on the coordinate axis. Find the length of the perpendicular from the origin on it.

Solution:

Since the plane makes intercepts −6, 3, 4, the equation becomes:

Let p be the distance of perpendicular from the origin to the plane.

⇒ 

⇒ 

⇒ 

⇒ 1/p² = 29/144 

⇒ p² = 144/29

⇒ p = 12/√29units.

Practice Problems on The Plane

Problem 1. Find the equation of the plane passing through the point (1, 2, 3) and perpendicular to the vector <2, -1, 4>.

Problem 2. Determine the distance of the plane 2x - 3y + 4z + 5 = 0 from the origin.

Problem 3. Find the angle between the planes 2x + y - z = 0 and x - 2y + 2z = 0.

Problem 4. Write the equation of the plane passing through the points (1, 0, 0), (0, 1, 0), and (0, 0, 1).

Problem 5. Find the equation of the plane that is parallel to the plane x + 2y - 3z + 4 = 0 and passes through the point (2, -1, 3).

Problem 6. Determine if the planes 2x + 3y - z = 5 and 4x + 6y - 2z = 10 are parallel.

Problem 7. Find the point of intersection of the planes x + y + z = 1, 2x - y + z = 2, and x + 2y - z = 3.

Problem 8. Calculate the distance between the parallel planes 2x + 3y - 6z + 7 = 0 and 2x + 3y - 6z - 5 = 0.

Problem 9. Find the equation of the plane that contains the line of intersection of the planes x + y + z = 1 and 2x - y + z = 0, and passes through the point (1, 2, -1).

Problem 10. Determine the image of the point (2, 3, 4) when reflected in the plane x + y + z = 0.

Also Read,

• Equation of plane
• Class 12 RD Sharma Mathematics Solutions - Chapter 29 The Plane - Exercise 29.6
• Class 12 RD Sharma Mathematics Solutions - Chapter 29 The Plane - Exercise 29.7
• Class 12 RD Sharma Mathematics Solutions - Chapter 29 The Plane - Exercise 29.8

Conclusion

Exercise 29.9 of RD Sharma's Class 12 Chapter 29 provides students with challenging problems involving the images of points with respect to planes and angle bisectors between planes. These solutions demonstrate the application of vector algebra and analytical geometry in solving complex spatial problems. By working through these problems, students develop a deeper understanding of the relationships between points and planes in three-dimensional space, as well as the concept of symmetry and reflection. These skills are fundamental in many areas of advanced mathematics, physics, and engineering, particularly in fields such as computer graphics, robotics, and structural analysis.