Exercise 12.2
1. Find the distance between the following pairs of points:
(i) (2, 3, 5) and (4, 3, 1)
(ii) 
and 
(iii) 
and 
(iv) 
and 
Ans. (i) Let A (2, 3, 5) and B (4, 3, 1) be two points, then
AB = 
= 
= 
units
(ii) Let A (–3, 7, 2) and B (2, 4, –1) be two points, then
AB = 
= 
= 
units
(iii) Let A (–1, 3, –4) and B (1, –3, 4) be two points, then
AB = 
= 
= 
units
(iv) Let A (2, –1, 3) and B (–2, 1, 3) be two points, then
AB = 
= 
= units
2. Show that the points 
and 
are collinear.
Ans. Let A (–2, 3, 5), B (1, 2, 3) and C (7, 0, –1) be three points, then
AB = 
= 
= 
units
BC = 
= 
= 
units
AC = = 
= 
= 
units
Here, AC = AB + BC
Therefore A, B and C are collinear.
3. Verify the following:
(i) 
and 
are the vertices of an isosceles triangle.
(ii) 
and 
are the vertices of right angled triangle.
(iii) 
and 
are the vertices of a parallelogram.
Ans. (i) Let A (0, 7, –10) B (1, 6, –6) and C (4, 9, –6) be three vertices of 
, then
AB = 
= 
= 
units
BC = 
= 
= units
AC = = 
= 
= 
units
Here, AB = BC
Therefore is an isosceles triangle.
(ii) Let A (–1, 2, 1), B (–1, 6, 6) and C (–4, 9, 6) be three vertices of , then
AB = 
= 
= 
units
BC = 
= 
= 
units
AC = = 
= 
= 
units
Here, AC2 = AB2 + BC2
Therefore is a right angled triangle.
(iii) Let A (0, 7, 10), B (1, –2, 5), C (4, –7, 8) and D (2, –3, 4) be four vertices of a quadrilateral ABCD, then
AB = 
= 
= 
units
BC = 
= 
= units
CD = 
= 
= units
AD = 
= 
= units
AC = 
= 
= 
units
BD = 
= 
= 
units
Here, AB = CD, BC = AD and AC 
BD
Therefore A, B, C and are the vertices of a parallelogram ABCD.
4. Find the equation of the set of points which are equidistant from the point (1, 2, 3) and 
Ans. Let A
be any point which is equidistant from points B (1, 2, 3) and C Then
According to question, AB = AC
Squaring both sides, we get
5. Find the equation of the set of points P, the sum of whose distance from A (4, 0, 0) and B
is equal to 10.
Ans. Let P
be any point, then
According to question, PA + PB = 10
Squaring both sides, we get
Again squaring both sides, we get
This is the required equation.
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