Exercise 26.2
Page no 26.12
Question 1
Find the distance between the following pair of points
(i) $(1,-3,4),(-4,1,2)$
(ii) $(-1,3,-4),(1,-3,4)$
(iii) $(2,3,4),(-1,2,3)$
(iv) $(2,3,5),(4,3,1)$
(v) $(0,1,-3),(3,0,5)$
(vi) $(2,-1,3),(-2,-1,3)$
(vii) $(-3,7,2),(2,4,-1)$
(viii) Find the coordinates of a point on $y$ axis which are at a distance of $5 \sqrt{2}$ from the point $P(3,-2,5)$.
Question 2
Show that the points $(0,7,-10),(1,6,-6)$ and $(4,9,-6)$ are the vertices of an isosceles triangle.
Question 3
(i) Prove that the points $(5,3,2),(3,2,5)$ and $(2,5,3)$ are the vertices of an equilateral triangle.
(ii) Show that the points $(a, b, c)(b, c, a),(c, a, b)$ are the vertices of an equilateral triangle.
Page no 26.13
(iii) Show that the points (0,7,10) (-4,9,6) are the vertices of a right angled triangle.
(iv) Are the points A(3,6,9) B( 10,20,30) AND C(25,-41,5) the vertices of a right angled triangle ?
Question 4
show that the points $(-2,3,5),(1,2,3)$ and $(7,0,-1)$ are collinear.
Question 5
Examine whether following points are collinear or not
(i) $(3,-2,4),(1,0,-2),(-1,2,-8)$
(ii) $(-3,7,2),(2,4,-1),(12,-2,-7)$
Question 6
Show that the points $P(-3,-2,4), Q(-9,-8,10)$ and $R(-5,-4,6)$ are collinear and $R$ divides $P Q$ in the ratio $1: 2$.
Question 7
Show that $(-1,4,-3)$ is the circumcenter of the triangle formed by the points $(3,2,-5),(-3,8,-5)$ and $(-3,2,1)$
Question 8
(i) Show that the points $(3,2,2),(-1,1,3),(0,5,6),(2,1,2)$ lie on a sphere whose centre is $(1,3,4)$. Find its radius.
(ii) Find the radius of the sphere through the points $(0,5,0),(4,3,0),(4,0,3)$ and $(0,4,3)$.
Question 9
Find the distance a from origin of the foot of perpendicular of point $(a, b, c)$ on xy-plane.
Question 10
Show that the coplanar points $(-1,2,1),(1,-2,5),(4,-7,8)$ and $(2,-3,4)$ are the vertices of a parallelogram.
Question 11
Show that the coplanar points $(-1,-6,10),(1,-3,4),(-5,-1,1)$ and $(-7,-4,7)$ are the vertices of a rhombus.
Question 12
Show that the coplanar points (1,5,3) (3,4,0) (5,6,1) and (3,7,3) are the vertices of a square .
Question 13
Examine whether the coplanar points (-2,6,-2) (0,4,-1) (-2,3,1) and (-4,5,0) are the vertices of a square .
Question 14
Find the point on y-axis which is equidistant from the points (5,5,2) and (3,1,2)
Question 15
Find the coordinates of the point equidistant from the points (0,0,0) (2,0,0)(0,4,0) and (0,0,6)
Question 16
Determine the point in $X Y$-plane which is equidistant from the points $A(1,-1,0), B(2,1,2)$ and $C(3,2,-1)$.
Question 17
Using distance formula, calculate the cosine of angle $A$ of the triangle with vertices $A(1,-1,2), B(6,11,2)$ and $C(1,2,6)$.
Question 18
(i) Find the locus of a point which moves so that its distances from the points $(3,4,-5)$ and $(-2,1,4)$ are equal.
(ii) Find the equation of the set of points which are equidistant from the points $(1,2,3)$ and $(3,2,-1)$
Question 19
If $P(-2,2,3)$ and $O(13,-3,13)$ are two points. Find the locus of point $R$ Which moves such that $3 P R=2 Q R$.
Question 20
Find the equation of the set of points $P$, the sum of whose distances from $A(<, 0,0)$ and $B(-4,0,0)$ is equal to $10 .$
Question 21
Find the locus of point P if $A P^{2}-B P^{2}=18$, where $A=(1,2,-3)$ and B=(3,-2,1)
Page no 26.14
Question 22
Find the equation of the set of points $P$ which moves so that its distances from the points $A(3,4,-5)$ and $B(-2,1,4)$ are equal.
Question 23
If $A$ and $B$ be the points $(3,4,5)$ and $(-1,3,-7)$ respectively, find the equation of set of points $P$ such that $P A^{2}+P B^{2}=k^{2}$, where $k$ is a constant.
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