KC Sinha Mathematics Solution Class 11 Chapter 26 Introduction to Three-Dimensional Geometry Exercise 26.3

 Exercise 26.3

Page no 26.24

Question 1

If $A(3,1,-2)$ and $B(1,-3,-1)$ be two points, find the coordinates of the poift which divides the line segment $A B$.

(i) internally in the ratio $1: 3$.

(ii) externally in the ratio 3:1.

Question 2

Find the coordinates of the point which divides the join of (-2,3,5)and  $(1,-4,-6)$ in the ratio

(i) $2: 3$ internally

(ii) $2: 3$ internally

Page no 26.25

Question 3

pod the coordinates of the point $R$ which divides $P Q$ externally in the ratio $2: 1$ ad verify that $Q$ is the mid-point of $P R$.

Question 4

Find the coordinates of the point R which divides the join of (0,0,0) and Q(4,-1,-2) in the 1:2 externally and verify that P is the mid point of RQ

Question 5

What section formula, show that the following three points are collinear :

(i) $(-2,3,5),(1,2,3),(7,0,-1)$

(ii) $(2,-1,3),(4,3,1),(3,1,2)$

(ii) $(-1,4,-2),(2,-2,1),(0,2,-1)$

(iv) $(2,3,4),(-1,-2,1),(5,8,7)$

(v) $(2,-3,4),(-1,2,1),\left(0, \frac{1}{3}, 2\right)$

(vi) $(-4,6,10),(2,4,6),(14,0,-2)$

Question 6

Find the coordinates of the points which trisect the line segment $P Q$ formed by joining the points $P(4,2,-6)$ and $Q(10,-16,6)$.

Question 7

Given that $P(3,2,-4), Q(5,4,-6)$ and $R(9,8,-10)$ are collinear. Find the ratio in which $Q$ divides $P R$.

Question 8

 Find the ratio in which the YZ-plane divides the line segment joining the following pair of points :

(i) $(4,8,10)$ and $(6,10,-8)$

(ii) $(-2,7,4)$ and $(3,-5,8)$.

Question 9

$A(3,2,0), B(5,3,2), C(-9,6,-3)$ are the vertices of $\triangle A B C$ and $A D$ is the bisector of $\angle B A C$ which meets $B C$ at $D$. Find the coordinates of $D$.

Question 10

Show that the points $(4,7,8),(2,3,4),(-1,-2,1)$ and $(1,2,5)$ are the vertices of a parallelogram.

Question 11

Prove that the points $(5,-1,1),(7,-4,7),(1,-6,10)$ and $(-1,-3,4)$ are the vertices of a rhombus.

Question 12

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 $A B C D$, but it is not a rectangle.

Question 13

(i) If three consecutive vertices of a parallelogram be $(3,4,-1),(7,10,-3)$ and $(8,1,0)$, find the fourth vertex.

(ii) Three vertices of a parallelogram $A B C D$ are $A(3,-1,2) B(1,2,-4)$ and $C(-1,1,2)$. Find the coordinates of the fourth vertex.

Question 14

Find the ratio in which the plane $3 x+4 y-5 z=1$ divides the line segment joining $(-2,4,-6)$ and $(3,-5,8)$.

Question 15

(i) A point $R$ with z-coordinates 8 lies on the line segment joining the points $P(2,-3,4)$ and $Q(8,0,10)$. Find the coordinates of $R$.

(ii) A point $R$ with x-coordinate 4 lies on the line segment join the points $P(2,-3,4)$ and $Q(8,0,10)$. Find the coordinates of $R$.

Question 16

Find the ratio in which the surface $x^{2}+y^{2}+z^{2}=504$ divides the line segment joining points $(12,-4,8)$ and $(27,-9,18)$.

Question 17

(i) Two vertices of a triangle are $(4,-6,3)$ and $(2,-2,1)$ and its centroid is $\left(\frac{8}{3},-1,2\right)$. Find the third vertex.

(ii) Find the lengths of the medians of the triangle having vertices $A(0,0,6)$ $B(0,4,0)$ and $C(6,0,0)$

Question 18

(i) If origin is the centroid of $\triangle A B C$ with vertices $A(\alpha, 1,3), B(-2, \beta,-5)$ and $C(4,7, \gamma)$. find the values of $\alpha, \beta$ and $\gamma$.

(ii) The origin is the centroid of the triangle $P Q R$ with vertices $P(2 a, 2,6)$, $Q(-4,3 b,-10)$ and $R(8,14,2 c)$ Then find the values of $a, b$ and $c$.

(iii) The centroid of a triangle $A B C$ is at the point $G(1,1,1)$ If the coordinates of $A$ and $B$ are $(3,-5,7)$ and $(-1,7,-6)$ respectively, then find the coordinates of the point $C$.

Question 19

Find the centroid of the triangle mid-points of whose sides are $(1,2,-3),(3,0,1)$ and $(-1,1,-4)$.

Question 20

If centroid of the tetrahedron $O A B C$, where coordinates of $A, B, C$ ate $(a, 2,3),(1, b, 2)$ and $(2,1, c)$ respectively be $(1,2,3)$, then find the distance of point $(a, b, c)$ from the origin.