The document discusses various aspects of geometry, focusing on the equations of lines, planes, and their relationships in space. It includes exercises on finding angles, distances, intersections, and properties of parallel and perpendicular lines and planes. Additionally, the document covers the conditions for coincident and rhombus formations, along with several geometric problems and propositions.

1. The angle between
t
h
e linesx=
2 andx-J3y+ l=0is-(30°,
60°, 120°, 150°)
4.
2
.
3
)
8 10
(
d
) The distance between
t
h
e lines 3x
1 =0 andx+3
=0is units.
(1, 4,
4,- 1)
(
2,3, -
2
,- 3)
Th
e lines 2r-3y +l=0
and3r
+ ky-l=0
are perpendicular
to each other
if k=
(0,- 1,3, -9)
(c) The lines
3r
+ ky 4= 0 and k-
4y- 3x =0 ar
e coincident
if k=
(
e
)
Th
e slope and
x - intercept
of t
h
e l
i
n
e 3x-y +k=
0 ar
e equal
if k=
EXERCISES
8 (b)
(
b
)
a)
Fi
llin th
e blanks
in each
of t
h
e following,
using t
h
e answersgiven againsteach
of them
:

2. 6 Obtain the cquation of straight lines
(a)) passing through (1, - 1) and having inclination 150°.
(b)) passing through (-1,2) and making intercept 2on the y- axis.
passing through the points (2. 3) and (4, ).
(d) passing through (-2. 3) and sum of whose intercepts ig 2.
1s0e,
distance from origin is 2 such that the perpendicular from
(e) whose perpena
origin has i
bisecting the line segment joining(3,4) and (1, 2) at right angles.

3. poruon between ne co-ordinate axes.
7(a)) Find the cquation of the lines which is parallell to the line 3.r + 4y +7=0 and is at
a distance 2 from it.
(b) Find the equations of the diagonals of the parallelogram formed by the lines ax +
by=0. ar + by +c=0, br + my = 0 and l + my + n=0. What is the condition that
this will be a rhombus ?
(c)} Find the equation of the line passing through the intersection of 2r-y-1 =0 and
3x- 4y + 6 =0 and parallel to the line x+ y-2=0.
d) Find the equation of theline passing through the point of intersection of lines x+
3y +2 = 0 and x-2y 4 =0 and perpendicular to the line 2y + 5x -9=0.
es) Find the equation of the line passing through intersection of the lines x +3y - | =0and
3x-y+ I=0 and the centroid of the triangle whose vertices are the points (3. -1).
(1,3) and (2. 4).

4. 6. Obtain the equation of straight lines :
(a)) passing through (1, 1) and having inclination 150°.
b))passing through (-1, 2) and making intercept 2 on the y- axis.
(c)
(d)
(e) whose perpendicular distance from origin is 2 such that the perpendicular from
origin has inclination 150°.
() bisecting the line segment joining (3,-4) and (1, 2)at right angles.
(h)
(g) bisecting the line segment joining (a, 0) and (0, b) at right angles.
biscting the line segments joining (a, b), (al, b') and (-a, b), (a', - b'),
passingthrough origin and the points of trisection of the portion of the line 3x +y
- 12= 0 intercepted between the co-ordinate axes.
(i)
()
passing through the points (2, 3) and (-4, 1)..
(k)
passing through (-2, 3) and sum of whose intercepts ig 2.
()
(b)
passing through (-4,2) and parallel to the line 4x - 3y=10.
passing through the point (a cos'0, a sin'e) and perpendicular to the straight line
Xseco+ y
cosece = a.
which passes through the point (3, - 4) and is such that its portion between the axes
is divided at this point internally in the ratio 2:3.
(m) which passes through the point (a, B) and is such that the given point bisects its
portion between the co-ordinate axes.
(a)) Findtheequation of the lines which is parallell to the line 3.r +4v+7=0and is at
a
distance 2 from it.
Findthe equations of the diagonals of the parallelogram formed by the lines ax +
by =0, ar + by +c=0,lx + my =0and x + my + n=0.What is the condition that
this will be a rhombus ?
((c)) Find the equation ofthe line passing through the intersection of 2r -y- l=0 and
3x 4y +6=0and parallel to the line x+y-2=0.
d) Find the equation of the line passing through the point of intersection of linesx+
3y +2 = 0 and x -2y 4=0and perpendicular to the line 2y + 5x 9=0.
Find the equation of the line passing through intersection ofthelines x+3y - l=0 and
3x-y + l=0and the centroid of the triangle whose vertices are the points (3, -1),
(1,3)and (2, 4).

5. Fillinthe blanks in each of thefollovwing questions by choosing the appropriate answer from the given
ones.
(a)
(a)
(b)
EXERCISE - 15 (a)
(c)
(b) The number of lines making equalangles with co-ordinate axes is - [one, two, four. eight)
(c) Ifa line isperpendicular to z-axis and makes an angle measuring 60° with x -axis then the angle
it makes with y - axis measures [30°, 60°, 90°, 120)
(d) The projection of the line segmentjoining(0,3, -1)and (3, 2,4)on zaxis is
(e) The image of the point (6. 3, 4) with respect to yz - plane is
(d)
The distance of the point P(x, y, z,) from z - axis is -
(e)
+ Zo
Ifthedistance between the points (-,-1, z) and (1, -, 1) is 2 then z =
()
Which of thefollowing statements are true (T) or false (F):
()
2 2
+y, +(Zo -2)
-.
[(6,0, -4), (6, -3, 4), (-6, -3, -4). (-6. 3. 4))
[1.V2, 2.o)
The line through (l-1,2) and (-2, -1,2)is always perpendicular to z-axis.
The line passingthrough.(0, 0, 0)and (1,2, 3) has directioncosines (-1,-2,-3).
1f.n, n be three realmumbers proportional to the direction cosines ofa line L. then f + m
n = |.
If a. B. y
be any three arbitrary angles then cos a., cos B, cos ycan always be considered as the
direction cosines ofa line.
There are four points in space which are at same distance from origin, as from (2. 3. -4)
rtwolines are perpendicular to athird line, then the direction ratios of thetwo lines are
proportional.
0,3.4.5]

6. (a)
(b)
IfA. BC are the points (1.4,2). (-2. 1.2) and (2. -3,4) respectivelythen find the angles ofthe
triangle ABC.
Find the acute angle between the lines passing through (-3, -1, 0). (2, -3, 1) and (1, 2, 3).
(z, 4. -2) respectively.
(c)/Prove that measure of the angle between two main diagonals of acube
Prove that measure of the angle betwenthe diagonal of aface and the diagönal of acube
drawn from avertex is cos
3)
Findthe angle which adiagonal ofacube makes with one of its edges.

7. (a)/ Eind the ratio in which the line sement through (I,3.-l) and(2. 6, -2) is divided by X- plane
(bind theratio in which thc lines segnentthrough(2.4,5).(3, 5,-4) is dividedby xy- planc.
(YFindtheco-ordinates ofthefoot of'theperpendicular fromthe point(I. 1, I)on the lincjoining
(d) Find the co-ordinates ofthepoint where theperpendicular fron the origin mects the line joining
he points (-9,4. 5) and (|1,0,-)
(0,4.6)and($. 4, 4).
(c) Find the co-ordinates of the centrojd ofthetriangle with its vertices at(a,. b,. c,). (a,. b,. c )and
sa,. b,,c).
()/A(|.0.-1), B(-2, 4.-2) andC0,5, 10) bethe vertices ofatrianglec andthe bisector of the
(g)
(h)
angle BAC, mcets BC atD. then lind the co-ordinates of the point D.
Prove that the points P(3, 2, 4), Q(5, 4, -6) and R(9, 8. -10) are collinear. Find the ratio in
which the point Qdivides the line segment PR.
Mind the ratio inwhich the line segment joining the points (2, -3, |). (3, -4, -5) is divided by the

8. State, which of thefollowving statements are true (T) or false ():
(a) Through any four points one and only one plane can pass.
(b) The equation of xy- plane is x+y=0.
(C) The plane ax + by + c=0 is perpendiculartoz - axis.
(a) The cquation oftheplane parallel to xZ-plane and passing through (2, -4, 0) is y +4 =0
(e) The planes 2x -y +z-l=0
and 6x - 3y +3z = |are coincident.
() The planes 2x+ 4y - z+|=0
and x- 2y- 6z + 3 =0 are perpendicular to each other.
(g) The distance of apoint from a plane is same as the distance of the point from any Iine lying in tha
plane.
|EXERCISE 15 (b)|
Fillinthe blanks by choosing the appropriate answer from the given ones :
(a) The equation ofa plane passingthrough (1, 1,2) andparallel to x+y+z-l=0 is
[x +y+z=0, X+y+2z -| =0
X+y+z=2, x+y+z=4]
(b) The equation of planeperpendicularto z- axis and passing through (1, -2, 4) is-
[x=1,y+2=0, z-4=0, X+y+z-3 =0]
(c) The distance between the parallel planes
2x-3y t 6z +|=0 and
4x 6yt 12z5=0 is
1 I 4 6
(d) The plane y - z+ |=0 is
[paralles tox-axis, perpendiculartox-axis, parallelto xy -plane, perpendicular toyz -plane.]
(e) Aplane whose normal has direction ratios <3,-2, k> is parallel to the line joining(-1, 1,4)and(5.t
-2). Then the value of k = [6, -4, -1, 0]
Findthe equation of planespassing through the points:
(a6,-1, ), (5, 1,2) and (1, -5,-4);
(6) (2,I, 3), (3. 2. 1) and (!, 0, -l);:
(c) (-1,0. 1),(-1,4, 2) and (2, 4, 1):
(d) (-1,5, 4). (2, 3, 4) and (2, 3, -);
(ey (1,2.3), (1,4, 3) and (-1, 3,2);
Find theequation ofplane in each ofthe following cases :
(a)-Passing through the point (2, 3,) and parallel to the plane 3x -4y +7z =0.
(b)_Passing throughthe points(2. -3, ) and(-l, 1,-7)and perpendiculartothe plane x-2y +Sz
+|=(
(a Passing through the foot ofthe perpendiculars drawn from P(a, b, c) on the coordinate planes.
(d) Passing through the point (-1,3,2) perpendicular totheplanes x+2y+2z =Sand 3x +3v+ 22 - S.
(e) Bisecting the line segment joining(-l,4, 3) and (5, -2,-1)at right angles.
Pafallel to theplane 2x - y
+3z+1=O and at adistance 3units away from it
a Writethe equation ofthe plane 3x - 4y+
-6z12 =0inintercept form and hence obtain the
co-ordinatc

9. 7
8.
of the pointswhere it mcets the co-ordinate axes.
(b)Write the cquation of theplane 2x -3y +5z +|=0 in normalform and find its distance from theorigin.
Find also the distacne trom the point (3. 1,2).
(c) Find thedistance between the parallei planes 2x- 2y + z+|=0 and 4x - 4y + 2z+ 3=0.
ln each the following cases, verify whether the four given points are coplanar or not.
a) (|,2. 3).(-1. I.0).(2. I, 3). (1. I.2)
(b) (1.I. I).(3. 1.2). (1.4.0). (-1, 1. 0)
(c) (0. -1,-).(4. 5. I).(3.9. 4),(-4. 4, 4)
(d) c6.3, 2).(3. -2, 4).(5, 7, 3). (-13. 17, -1)
Find the equation ofplane in cach ofthefollowing cases:
(a) Passing through the intersection of planes 2x + 3y -4z +| =0and 3x - y+z+2=0 and passing
through the point (3, 2.|).
(b) Which contains the line of intersection of the planes x + 2y + 32 - 4 = 0, 2x + y -z +5 =0 and
perpendicular to the plane 5x +3y +6z +8=0.
(c) Passingthrough the intersection ofax+by +cz +d=0 and a,x +b,y +c,z+d,=0and perpendicular
toxy - planc.
(d) Passing through the intersection oftheplanes x+3y - z+l=0 and 3x - y+Sz+3 =0 and is at a
2
distance unitsfrom origin.
3
find the angle betwecn the fol!owing pairs ofplanes.
Aa) x+3v- 52.+
(b) x + 2y + 2z -
|=0and 2x +y- z+3 0
3=0and 3x + 4y +Sz+|=0
(c) x 2y + 22- 7=0and 2x - y t z=6

10. J
4.
Findthe centre and radius fthefollowing spheres :
(a)
(b)
(c)
(d)
(a)
(b)
)
Find the cquation of spheres in each ofthefollowing cases :
(d)
(e)
()
(h)
3x+3v+ 3z°- 12x -6v + 9z + | =0:;
(i)
7x + 7y+ 77 -6x - 3y - 2z =0:
X*+y tz -4X + 2y - 22- 10=0:
UN + yt uz + 2ux + 2vy + 2wz +d= 0
Centre at (3. 1,-2) and sphere passingthrough(1. I, 2):
Centre at (2. -1.4) and the sphere touches the plane 2x - y- 27 +6=0.
Centre at origin and sphere touchesthc line 2(x + )=(2-y) =(z +3:
Passingthroughthe points (0.0,0).(0, I, -).(-1, 2. 0) and(0.2.3):
Passing through the points (0.0. 0).(-u, b,c). (u. -b. c) and (u, b. -c):
Passingthrough(0.0. 1). (1, 0. 0). (0. 1.0) and louching theplane 2x +2y-z= !
Passing through (0. -2. -4), (2, -1. )andcentre lies on the line 5x+ 22 = ) 2x
Passing through (-1, 6. 6).(0. 7. 10)4-4,9,6)andcentre lying onthe plane 2x+ 2
Passing through (1, 2, -3) and (3. -1,2) and centre lying on X-axis.
(ý) Passing through (4, 5. -6) and centre being the point of intersection