2D_circle & 3D_sphere.all question for class 12th

Co ordinate geometry question for class 12th

1. The angle between the lines x= 2 andx-J3y+ l=0is-(30°, 60°, 120°, 150°) (4.2.3)
8 10
(d) The distance between the lines 3x 1 =0 andx+3 =0 is units.
(1, 4, 4, -1)
(2, 3, -2, -3)
The lines 2r-3y +l=0 and 3r + 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 are coincident if k=
(e)
The slope and x -intercept of the line 3x-y +k= 0 are equal if k=
EXERCISES 8 (b)
(b)
a)
Fill in the blanks in each of the following, using the answers given against each 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 2 on 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 the line 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'),
passing through 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
X seco+ 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)) Find the equation of the lines which is parallell to the line 3.r + 4v+7=0and is at
a distance 2 from it.
Find the equations of the diagonals of the parallelogram formed by the lines ax +
by = 0, ar + by +c=0, lx + my =0 and x + 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-l =0 and 3x 4y +6=0 and 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 =0 and
perpendicular to the line 2y + 5x 9=0. Find the equation of the line passing through intersection of the lines x +3y -l=0 and 3x-y + l=0 and the centroid of the triangle whose vertices are the points (3, -1), (1,3) and (2, 4).

5. Fill in the blanks in each of the follovwing questions by choosing the appropriate answer from the given
ones.
(a)
(a)
(b)
EXERCISE -15 (a)
(c)
(b) The number of lines making equal angles with co-ordinate axes is -[ one, two, four. eight)
(c) Ifa line is perpendicular 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 segment joining (0,3, -1) and (3, 2,4) on z axis 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
If the distance between the points (-,-1, z) and (1, -, 1) is 2 then z =
()
Which of the following 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 passing through. (0, 0, 0) and (1,2, 3) has direction cosines (-1,-2,-3).
1f. n, n be three real mumbers 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 y can 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)
r two lines are perpendicular to a third line, then the direction ratios of the two 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) respectively then find the angles of the
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 a cube
Prove that measure of the angle betwen the diagonal of a face and the diagönal of a cube
drawn from a vertex is cos 3)
Find the angle which a diagonal of a cube 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 the ratio in which thc lines segnent through(2.4, 5).(3, 5,-4) is divided by xy-planc.
(YFind the co-ordinates of the foot of' the perpendicular from the point (I. 1, I) on the linc joining
(d) Find the co-ordinates of the point where the perpendicular 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 of the triangle with its vertices at(a,. b,. c,). (a,. b,. c ) and
sa,. b,, c).
()/A(|.0.-1), B(-2, 4. -2) andC0,5, 10) be the vertices ofa trianglec and the bisector of the
(g)
(h)
angle BAC, mcets BC at D. 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 Q divides the line segment PR.
Mind the ratio in which the line segment joining the points (2, -3, |). (3, -4, -5) is divided by the

8. State, which of the followving 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 perpendicular to z -axis.
(a) The cquation of the plane 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 a point from a plane is same as the distance of the point from any Iine lying in tha
plane.
|EXERCISE 15 (b)|
Fill in the blanks by choosing the appropriate answer from the given ones :
(a) The equation ofa plane passing through (1, 1,2) and parallel 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 plane perpendicular to 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, perpendicular tox -axis, parallel to xy -plane, perpendicular to yz -plane.]
(e) A plane 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]
Find the equation of planes passing 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 the equation of plane in each of the following cases :
(a)-Passing through the point (2, 3, ) and parallel to the plane 3x -4y + 7z = 0.
(b)_Passing through the points (2. -3, ) and (-l, 1,-7) and perpendicular to the plane x-2y +
Sz
+|=(
(a Passing through the foot of the perpendiculars drawn from P (a, b, c) on the coordinate planes.
(d) Passing through the point (-1,3,2) perpendicular tothe planes 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 the plane 2x -y + 3z+1=O and at a distance 3 units away from it
a Write the equation of the plane 3x -4y+ -6z12 =0 in intercept form and hence obtain the
co-ordinatc

9. 7
8.
of the points where it mcets the co-ordinate axes.
(b) Write the cquation of the plane 2x -3y + 5z + | =0 in normal form and find its distance from the origin.
Find also the distacne trom the point (3. 1,2).
(c) Find the distance 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 of plane in cach of the following cases:
(a) Passing through the intersection of planes 2x + 3y -4z + | =0 and 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) Passing through the intersection of ax + by + cz +d=0 and a,x + b,y + c,z+d, =0 and perpendicular
to xy -planc.
(d) Passing through the intersection of the planes x + 3y -z+l =0 and 3x -y + Sz+3 =0 and is at a
2
distance units from origin.
3
find the angle betwecn the fol!owing pairs of planes.
Aa) x + 3v-52.+
(b) x + 2y + 2z -
|=0 and 2x + y-z +3 0
3=0 and 3x + 4y + Sz+|=0
(c) x 2y + 22-7=0 and 2x -y t z=6

10. J
4.
Find the centre and radius f the following spheres :
(a)
(b)
(c)
(d)
T
(a)
(b)
)
Find the cquation of spheres in each of the following 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 passing through (1. I, 2):
Centre at (2. -1.4) and the sphere touches the plane 2x -y-27 +6=0.
Centre at origin and sphere touches thc line 2(x + )=(2-y) = (z + 3:
Passing through the 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):
Passing through (0.0. 1). (1, 0. 0). (0. 1.0) and louching the plane 2x +2y-z= !
Passing through (0. -2. -4), (2, -1. ) and centre lies on the line 5x+ 22 = ) 2x
Passing through (-1, 6. 6).(0. 7. 10)4-4,9,6) and centre lying on the 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