Chapters
Chapter 2: Relations
Chapter 3: Functions
Chapter 4: Measurement of Angles
Chapter 5: Trigonometric Functions
Chapter 6: Graphs of Trigonometric Functions
Chapter 7: Values of Trigonometric function at sum or difference of angles
Chapter 8: Transformation formulae
Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle
Chapter 10: Sine and cosine formulae and their applications
Chapter 11: Trigonometric equations
Chapter 12: Mathematical Induction
Chapter 13: Complex Numbers
Chapter 14: Quadratic Equations
Chapter 15: Linear Inequations
Chapter 16: Permutations
Chapter 17: Combinations
Chapter 18: Binomial Theorem
Chapter 19: Arithmetic Progression
Chapter 20: Geometric Progression
Chapter 21: Some special series
Chapter 22: Brief review of cartesian system of rectangular co-ordinates
Chapter 23: The straight lines
Chapter 24: The circle
Chapter 25: Parabola
Chapter 26: Ellipse
Chapter 27: Hyperbola
Chapter 28: Introduction to three dimensional coordinate geometry
Chapter 29: Limits
Chapter 30: Derivatives
Chapter 31: Mathematical reasoning
Chapter 32: Statistics
Chapter 33: Probability
Chapter 28: Introduction to three dimensional coordinate geometry
RD Sharma solutions for Class 11 Mathematics Textbook Chapter 28 Introduction to three dimensional coordinate geometry Exercise 28.1 [Pages 6 - 7]
Name the octants in which the following points lie:
(–5, 4, 3)
Name the octants in which the following points lie:
(4, –3, 5)
Name the octants in which the following points lie:
(7, 4, –3)
Name the octants in which the following points lie:
(–5, –4, 7)
Name the octants in which the following points lie:
(–5, –3, –2)
Name the octants in which the following points lie:
(2, –5, –7)
Name the octants in which the following points lie:
(–7, 2 – 5)
Find the image of:
(–2, 3, 4) in the yz-plane.
Find the image of:
(–5, 4, –3) in the xz-plane.
Find the image of:
(5, 2, –7) in the xy-plane.
Find the image of:
(–5, 0, 3) in the xz-plane.
Find the image of:
(–4, 0, 0) in the xy-plane.
A cube of side 5 has one vertex at the point (1, 0, –1), and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the coordinates of the other vertices of the cube.
Planes are drawn parallel to the coordinate planes through the points (3, 0, –1) and (–2, 5, 4). Find the lengths of the edges of the parallelepiped so formed.
Planes are drawn through the points (5, 0, 2) and (3, –2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed.
Find the distances of the point P(–4, 3, 5) from the coordinate axes.
The coordinates of a point are (3, –2, 5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point.
RD Sharma solutions for Class 11 Mathematics Textbook Chapter 28 Introduction to three dimensional coordinate geometry Exercise 28.2 [Pages 9 - 10]
Find the distance between the following pairs of points:
P(1, –1, 0) and Q(2, 1, 2)
Find the distance between the following pairs of point:
A(3, 2, –1) and B(–1, –1, –1).
Find the distance between the points P and Q having coordinates (–2, 3, 1) and (2, 1, 2).
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Using distance formula prove that the following points are collinear:
A(4, –3, –1), B(5, –7, 6) and C(3, 1, –8)
Using distance formula prove that the following points are collinear:
P(0, 7, –7), Q(1, 4, –5) and R(–1, 10, –9)
Using distance formula prove that the following points are collinear:
A(3, –5, 1), B(–1, 0, 8) and C(7, –10, –6)
Determine the points in xy-plan are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
Determine the points in yz-plane and are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
Determine the points in zx-plane are equidistant from the points A(1, –1, 0), B(2, 1, 2) and C(3, 2, –1).
Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, –4).
Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).
Find the points on z-axis which are at a distance \[\sqrt{21}\]from the point (1, 2, 3).
Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle.
Show that the points (0, 7, 10), (–1, 6, 6) and (–4, 9, 6) are the vertices of an isosceles right-angled triangle.
Show that the points A(3, 3, 3), B(0, 6, 3), C(1, 7, 7) and D(4, 4, 7) are the vertices of a square.
Prove that the point A(1, 3, 0), B(–5, 5, 2), C(–9, –1, 2) and D(–3, –3, 0) taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.
Show that the points A(1, 3, 4), B(–1, 6, 10), C(–7, 4, 7) and D(–5, 1, 1) are the vertices of a rhombus.
Prove that the tetrahedron with vertices at the points O(0, 0, 0), A(0, 1, 1), B(1, 0, 1) and C(1, 1, 0) is a regular one.
Show that the points (3, 2, 2), (–1, 4, 2), (0, 5, 6), (2, 1, 2) lie on a sphere whose centre is (1, 3, 4). Find also its radius.
Find the coordinates of the point which is equidistant from the four points O(0, 0, 0), A(2, 0, 0), B(0, 3, 0) and C(0, 0, 8).
If A(–2, 2, 3) and B(13, –3, 13) are two points.
Find the locus of a point P which moves in such a way the 3PA = 2PB.
Find the locus of P if PA2 + PB2 = 2k2, where A and B are the points (3, 4, 5) and (–1, 3, –7).
Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.
Are the points A(3, 6, 9), B(10, 20, 30) and C(25, –41, 5), the vertices of a right-angled triangle?
Verify the following:
(0, 7, –10), (1, 6, –6) and (4, 9, –6) are vertices of an isosceles triangle.
Verify the following:
(0, 7, 10), (–1, 6, 6) and (–4, 9, –6) are vertices of a right-angled triangle.
Verify the following:
(–1, 2, 1), (1, –2, 5), (4, –7, 8) and (2, –3, 4) are vertices of a parallelogram.
Verify the following:
(5, –1, 1), (7, –4,7), (1, –6,10) and (–1, – 3,4) are the vertices of a rhombus.
Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, –1).
Find the locus of the point, the sum of whose distances from the points A(4, 0, 0) and B(–4, 0, 0) is equal to 10.
Show that the points A(1, 2, 3), B(–1, –2, –1), C(2, 3, 2) and D(4, 7, 6) are the vertices of a parallelogram ABCD, but not a rectangle.
Find the equation of the set of the points P such that its distances from the points A(3, 4, –5) and B(–2, 1, 4) are equal.
RD Sharma solutions for Class 11 Mathematics Textbook Chapter 28 Introduction to three dimensional coordinate geometry Exercise 28.3 [Pages 19 - 20]
The vertices of the triangle are A(5, 4, 6), B(1, –1, 3) and C(4, 3, 2). The internal bisector of angle A meets BC at D. Find the coordinates of D and the length AD.
A point C with z-coordinate 8 lies on the line segment joining the points A(2, –3, 4) and B(8, 0, 10). Find its coordinates.
Show that the three points A(2, 3, 4), B(–1, 2 – 3) and C(–4, 1, –10) are collinear and find the ratio in which C divides AB.
Find the ratio in which the line joining (2, 4, 5) and (3, 5, 4) is divided by the yz-plane.
Find the ratio in which the line segment joining the points (2, –1, 3) and (–1, 2, 1) is divided by the plane x + y + z = 5.
If the points A(3, 2, –4), B(9, 8, –10) and C(5, 4, –6) are collinear, find the ratio in which Cdivides AB.
The mid-points of the sides of a triangle ABC are given by (–2, 3, 5), (4, –1, 7) and (6, 5, 3). Find the coordinates of A, B and C.
A(1, 2, 3), B(0, 4, 1), C(–1, –1, –3) are the vertices of a triangle ABC. Find the point in which the bisector of the angle ∠BAC meets BC.
Find the ratio in which the sphere x2 + y2 + z2 = 504 divides the line joining the points (12, –4, 8) and (27, –9, 18).
Show that the plane ax + by + cz + d = 0 divides the line joining the points (x1, y1, z1) and (x2, y2, z2) in the ratio \[- \frac{a x_1 + b y_1 + c z_1 + d}{a x_2 + b y_2 + c z_2 + d}\]
Find the centroid of a triangle, mid-points of whose sides are (1, 2, –3), (3, 0, 1) and (–1, 1, –4).
The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, –5, 7) and (–1, 7, –6) respectively, find the coordinates of the point C.
Find the coordinates of the points which tisect the line segment joining the points P(4, 2, –6) and Q(10, –16, 6).
Using section formula, show that he points A(2, –3, 4), B(–1, 2, 1) and C(0, 1/3, 2) are collinear.
Given that P(3, 2, –4), Q(5, 4, –6) and R(9, 8, –10) are collinear. Find the ratio in which Qdivides PR.
Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, –8) is divided by the yz-plane.
RD Sharma solutions for Class 11 Mathematics Textbook Chapter 28 Introduction to three dimensional coordinate geometry [Page 22]
Write the distance of the point P (2, 3,5) from the xy-plane.
Write the distance of the point P(3, 4, 5) from z-axis.
If the distance between the points P(a, 2, 1) and Q (1, −1, 1) is 5 units, find the value of a.
The coordinates of the mid-points of sides AB, BC and CA of △ABC are D(1, 2, −3), E(3, 0,1) and F(−1, 1, −4) respectively. Write the coordinates of its centroid.
Write the coordinates of the foot of the perpendicular from the point (1, 2, 3) on y-axis.
Write the length of the perpendicular drawn from the point P(3, 5, 12) on x-axis.
Write the coordinates of third vertex of a triangle having centroid at the origin and two vertices at (3, −5, 7) and (3, 0, 1).
What is the locus of a point for which y = 0, z = 0?
Find the ratio in which the line segment joining the points (2, 4,5) and (3, −5, 4) is divided by the yz-plane.
Find the point on y-axis which is at a distance of \[\sqrt{10}\] units from the point (1, 2, 3).
Find the point on x-axis which is equidistant from the points A (3, 2, 2) and B (5, 5, 4).
Find the coordinates of a point equidistant from the origin and points A (a, 0, 0), B (0, b, 0) andC(0, 0, c).
Write the coordinates of the point P which is five-sixth of the way from A(−2, 0, 6) to B(10, −6, −12).
If a parallelopiped is formed by the planes drawn through the points (2,3,5) and (5, 9, 7) parallel to the coordinate planes, then write the lengths of edges of the parallelopiped and length of the diagonal.
Determine the point on yz-plane which is equidistant from points A(2, 0, 3), B(0, 3,2) and C(0, 0,1).
If the origin is the centroid of a triangle ABC having vertices A(a, 1, 3), B(−2, b −5) and C (4, 7, c), find the values of a, b, c.
RD Sharma solutions for Class 11 Mathematics Textbook Chapter 28 Introduction to three dimensional coordinate geometry [Pages 22 - 23]
The ratio in which the line joining (2, 4, 5) and (3, 5, –9) is divided by the yz-plane is
2 : 3
3 : 2
–2 : 3
4 : –3
The ratio in which the line joining the points (a, b, c) and (–a, –c, –b) is divided by the xy-plane is
a : b
b : c
c : a
c : b
If P(0, 1, 2), Q(4, –2, 1) and O(0, 0, 0) are three points, then ∠POQ =
\[\frac{\pi}{6}\]
\[\frac{\pi}{4}\]
\[\frac{\pi}{3}\]
\[\frac{\pi}{2}\]
If the extremities of the diagonal of a square are (1, –2, 3 and (2, –3, 5), then the length of the side is
\[\sqrt{6}\]
\[\sqrt{3}\]
\[\sqrt{5}\]
\[\sqrt{7}\]
The points (5, –4, 2), (4, –3, 1), (7, 6, 4) and (8, –7, 5) are the vertices of
a rectangle
a square
a parallelogram
none of these
In a three dimensional space the equation x2 – 5x + 6 = 0 represents
points
planes
curves
pair of straight lines
Let (3, 4, –1) and (–1, 2, 3) be the end points of a diameter of a sphere. Then, the radius of the sphere is equal to
2
3
6
7
XOZ-plane divides the join of (2, 3, 1) and (6, 7, 1) in the ratio
3 : 7
2 : 7
–3 : 7
–2 : 7
What is the locus of a point for which y = 0, z = 0?
x - axis
y - axis
z - axis
yz - plane
The coordinates of the foot of the perpendicular drawn from the point P(3, 4, 5) on the yz- plane are
(3, 4, 0)
(0, 4, 5)
(3, 0, 5)
(3, 0, 0)
The coordinates of the foot of the perpendicular from a point P(6,7, 8) on x - axis are
(6, 0, 0)
(0, 7, 0)
(0, 0, 8)
(0, 7, 8)
We know that the y and z coordinates on x - axis are 0
The coordinates of the foot of the perpendicular from a point P(6,7, 8) on x - axis are (6, 0, 0)
Hence, the correct answer is option (a).
The perpendicular distance of the point P (6, 7, 8) from xy - plane is
8
7
6
10
The length of the perpendicular drawn from the point P (3, 4, 5) on y-axis is
10
\[\sqrt{34}\]
\[\sqrt{113}\]
512
The perpendicular distance of the point P(3, 3,4) from the x-axis is
\[3\sqrt{2}\]
5
3
4
The length of the perpendicular drawn from the point P(a, b, c) from z-axis is
\[\sqrt{a^2 + b^2}\]
\[\sqrt{b^2 + c^2}\]
\[\sqrt{a^2 + c^2}\]
\[\sqrt{a^2 + b^2 + c^2}\]
Chapter 28: Introduction to three dimensional coordinate geometry
RD Sharma solutions for Class 11 Mathematics Textbook chapter 28 - Introduction to three dimensional coordinate geometry
RD Sharma solutions for Class 11 Mathematics Textbook chapter 28 (Introduction to three dimensional coordinate geometry) 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 11 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.
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Concepts covered in Class 11 Mathematics Textbook chapter 28 Introduction to three dimensional coordinate geometry are Coordinate Axes and Coordinate planes, Coordinates of a Point in Space, Distance Between Two Points, Section Formula, Three Dimessional Space.
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