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Title: Introduction To 3D Dimensional Geometry
Description: In this file you will Get notes of introduction To 3 Dimensional Geometry Of Mathematics
Description: In this file you will Get notes of introduction To 3 Dimensional Geometry Of Mathematics
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NCERT Solutions for Class 11 Maths Chapter 12
Introduction to Three Dimensional Geometry Class 11
Chapter 12 Introduction to Three Dimensional Geometry Exercise 12
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2, 12
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1 : Solutions of Questions on Page Number : 271
Q1 :
A point is on the x-axis
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Q2 :
A point is in the XZ-plane
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Q3 :
Name the octants in which the following points lie:
(1, 2, 3), (4, -2, 3), (4, -2, -5), (4, 2, -5), (-4, 2, -5), (-4, 2, 5),
(-3, -1, 6), (2, -4, -7)
Answer :
The x-coordinate, y-coordinate, and z-coordinate of point (1, 2, 3) are all positive
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The x-coordinate, y-coordinate, and z-coordinate of point (4, -2, 3) are positive, negative, and positive respectively
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The x-coordinate, y-coordinate, and z-coordinate of point (4, -2, -5) are positive, negative, and negative respectively
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The x-coordinate, y-coordinate, and z-coordinate of point (4, 2, -5) are positive, positive, and negative respectively
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The x-coordinate, y-coordinate, and z-coordinate of point (-4, 2, -5) are negative, positive, and negative respectively
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The x-coordinate, y-coordinate, and z-coordinate of point (-4, 2, 5) are negative, positive, and positive respectively
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The x-coordinate, y-coordinate, and z-coordinate of point (-3, -1, 6) are negative, negative, and positive respectively
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The x-coordinate, y-coordinate, and z-coordinate of point (2, -4, -7) are positive, negative, and negative respectively
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Q4 :
Fill in the blanks:
Answer :
(i) The x-axis and y-axis taken together determine a plane known as
(ii) The coordinates of points in the XY-plane are of the form
(iii) Coordinate planes divide the space into
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octants
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2 : Solutions of Questions on Page Number : 273
Q1 :
Find the distance between the following pairs of points:
(i) (2, 3, 5) and (4, 3, 1) (ii) (-3, 7, 2) and (2, 4, -1)
(iii) (-1, 3, -4) and (1, -3, 4) (iv) (2, -1, 3) and (-2, 1, 3)
Answer :
The distance between points P(x1, y1, z1) and P(x2, y2, z2) is given
by
(i) Distance between points (2, 3, 5) and (4, 3, 1)
(ii) Distance between points (–3, 7, 2) and (2, 4, –1)
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(iii) Distance between points (–1, 3, –4) and (1, –3, 4)
(iv) Distance between points (2, –1, 3) and (–2, 1, 3)
Q2 :
Show that the points (-2, 3, 5), (1, 2, 3) and (7, 0, -1) are collinear
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Points P, Q, and R are collinear if they lie on a line
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Q3 :
Verify the following:
(i) (0, 7, -10), (1, 6, -6) and (4, 9, -6) are the vertices of an isosceles triangle
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(iii) (-1, 2, 1), (1, -2, 5), (4, -7, 8) and (2, -3, 4) are the vertices of a parallelogram
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Here, AB = BC ≠ CA
Thus, the given points are the vertices of an isosceles triangle
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Therefore, by Pythagoras theorem, ABC is a right triangle
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(iii) Let (–1, 2, 1), (1, –2, 5), (4, –7, 8), and (2, –3, 4) be denoted by A, B, C, and D respectively
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Therefore, ABCD is a parallelogram
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Q4 :
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Find the equation of the set of points which are equidistant from the points (1, 2, 3) and (3, 2, -1)
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Accordingly, PA = PB
⇒ x2 – 2x + 1 + y2 – 4y + 4 + z2 – 6z + 9 = x2 – 6x + 9 + y2 – 4y + 4 + z2 + 2z + 1
⇒ –2x –4y – 6z + 14 = –6x – 4y + 2z + 14
⇒ – 2x – 6z + 6x – 2z = 0
⇒ 4x – 8z = 0
⇒ x – 2z = 0
Thus, the required equation is x – 2z = 0
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Answer :
Let the coordinates of P be (x, y, z)
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It is given that PA + PB = 10
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Exercise 12
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Answer :
(i) The coordinates of point R that divides the line segment joining points P (x1, y1, z1) and Q (x2, y2, z2) internally in the
ratio m: n are
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(ii) The coordinates of point R that divides the line segment joining points P (x1, y1, z1) and Q (x2, y2, z2) externally in
the ratio m: n are
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Q2 :
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Given that P (3, 2, -4), Q (5, 4, -6) and R (9, 8, -10) are collinear
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Answer :
Let point Q (5, 4, –6) divide the line segment joining points P (3, 2, –4) and R (9, 8, –10) in the ratio k:1
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Q3 :
Find the ratio in which the YZ-plane divides the line segment formed by joining the points (-2, 4, 7) and (3, -5,
8)
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Hence, by section formula, the coordinates of point of intersection are given by
On the YZ plane, the x-coordinate of any point is zero
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Q4 :
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Using section formula, show that the points A (2, –3, 4), B (–1, 2, 1) and
are collinear
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Let P be a point that divides AB in the ratio k:1
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By taking
, we obtain k = 2
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is a point that divides AB externally in the ratio 2:1 and is the same as point P
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Q5 :
Find the coordinates of the points which trisect the line segment joining the points P (4, 2, -6) and Q (10, -16,
6)
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Therefore, by section formula, the coordinates of point A are given by
Point B divides PQ in the ratio 2:1
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Exercise Miscellaneous : Solutions of Questions on Page Number : 278
Q1 :
Three vertices of a parallelogram ABCD are A (3, -1, 2), B (1, 2, -4) andC (-1, 1, 2)
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Answer :
The three vertices of a parallelogram ABCD are given as A (3, –1, 2), B (1, 2, –4), and C (–1, 1, 2)
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We know that the diagonals of a parallelogram bisect each other
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∴Mid-point of AC = Mid-point of BD
⇒ x = 1, y = –2, and z = 8
Thus, the coordinates of the fourth vertex are (1, –2, 8)
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Answer :
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Let AD, BE, and CF be the medians of the given triangle ABC
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∴Coordinates of point D =
Thus, the lengths of the medians of ΔABC are
= (3, 2, 0)
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Answer :
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It is known that the coordinates of the centroid of the triangle, whose vertices are (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3),
are
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Thus, the respective values of a, b, and c are
Q4 :
Find the coordinates of a point on y-axis which are at a distance of
from the point P (3, –2, 5)
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Let A (0, b, 0) be the point on the y-axis at a distance of
from point P (3, –2, 5)
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Q5 :
A point R with x-coordinate 4 lies on the line segment joining the pointsP (2, –3, 4) and Q (8, 0, 10)
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[Hint suppose R divides PQ in the ratio k: 1
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Let R divide line segment PQ in the ratio k:1
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Therefore, the coordinates of point R are
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Q6 :
If A and B be the points (3, 4, 5) and (-1, 3, -7), respectively, find the equation of the set of points P such that
PA2 + PB2 = k2, where k is a constant
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Let the coordinates of point P be (x, y, z)
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Title: Introduction To 3D Dimensional Geometry
Description: In this file you will Get notes of introduction To 3 Dimensional Geometry Of Mathematics
Description: In this file you will Get notes of introduction To 3 Dimensional Geometry Of Mathematics