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UNIT II
THREE DIMENSIONAL GEOMETRY
PART – A
1) Find the volume of the sphere
x + y + z 2 + 2x − 4 y + 8z − 2 = 0
2
2
ANS
Radius of the sphere r = 1 + 4 + 16 + 2
=
Volume of the sphere =
23
4 3 92π 23
πr =
3
3
2) Show that the plane 2x-2y+z+12=0 touches the sphere
x2 + y 2 + z 2 − 2 x − 4 y + 2z − 3 = 0
ANS
Radius =
u 2 + v 2 + w2 − d =
6+3 = 3
Perpendicular distance from thje center (1, 2, -1) to the plane
2x-2y+z+12=0 =
=
±
2 − 4 − 1 + 12
4 + 4 +1
9
=3
3
∴ The plane touches the sphere.
3) Find the equation of the sphere concentric with
x 2 + y 2 + z 2 − 2 x − 2 y − 2 z − 1 = 0 and passing
through the point (-2 ,1 ,5).
ANS
The equation of the sphere is
x 2 + y 2 + z 2 − 2 x − 2 y − 2 z + k = 0 -------------( 1 )
Sphere (1) passes through the point (-2, 1, 5)
∴ 4+1+25+4-2-10+k = 0
⇒ k= - 22
∴ The required sphere is x 2 + y 2 + z 2 − 2 x − 2 y − 2 z − 22 = 0
4) The point (2 ,3 ,4) is one end of the diameter of a sphere
other end.
ANS
Centre of the sphere = (1 ,1, -2)
If the other end is the point ( x ,y ,z), then
x+2
y +3
z+4
= 1,
= 1 and
= −2
2
2
2
⇒ x = 0, y = -1,
z = -8
∴ The other end is ( 0, -1, -8)
x 2 + y 2 + z 2 − 2 x − 2 y + 4 z − 1 = 0 ,find the
5) Find the equation of the normal at the point (2,-1,4) to the sphere
ANS
x 2 + y 2 + z 2 − y − 2 z − 14 = 0 .
The equation of the tangent plane at (2,-1,4) is
x(2)+y(-1)+z(4)-
1
(y-1)-(z+4)-14= 0
2
4x-2y+8z-y+1-2z-8-28=0
4x-3y+6z-35=0
Therefore Equation of the normal is
x − 2 y +1 z − 4
=
=
4
−3
6
6) Write down the equation of the sphere whose diameter is the line joining (1,1,1) and
(-1,-1,-1).
ANS
The equation of the sphere is
(x-1)(x+1)+(y-1)(y+1)+(z-1)(z+1)=0
x2 + y 2 + z 2 = 3
7) Find the equation of the sphere with centre at (2,3,5) which touches the XOY-plane.
ANS
Centre = (2,3,5) and radius = diatance between (2,3,5) and (2,3,0)
∴ Radius = 5
The equation of the sphere is
( x − 2) 2 + ( y − 3) 2 + ( z − 5) 2 = 52
∴ x 2 + y 2 + z 2 − 4 x − 6 y − 10 z + 13 = 0
8) Find the equation of the sphere whose centre (2,-3,4) and radius 5.
ANS
( x − 2) 2 + ( y + 3) 2 + ( z − 5) 2 = 52
x 2 + y 2 + z 2 − 4 x + 6 y − 8 z + 4 = 0 is the required sphere.
9) Find the centre and radius of the sphere
ANS
7 x 2 + 7 y 2 + 7 z 2 + 28 x − 42 y + 56 z + 3 = 0 .
a =7, u =14, v = -21, w = 28, d = 3.
Centre =
1 2 2
 u v w
u + v + w2 − ad
 − , − , −  and Radius =
a
 a a a
Centre = (-2,3,-4) and Radius =
1
196 + 441 + 784 − 21
7
Radius =
1400
2
= 10
.
7
7
10)Find the equation of the cone with vertex at the origin and which passes through the curve
x2 + y2 = 4,z=2
ANS
x 2 + y 2 = 4 and z =2 gives
x2 + y 2 = z 2
x 2 + y 2 − z 2 = 0 is the required cone.
11) Find the equation of the cone which passes through the curve x 2 + y 2 + z 2 = 9 ,x + y + z =1
and whose vertex is at the origin.
ANS
The equation of the cone is given by
x 2 + y 2 + z 2 = 9( x + y + z ) 2
⇒ 4 ( x 2 + y 2 + z 2 ) + 9( xy + yz + zx) = 0
12) Find the equation of the cone whose vertex is the point (1,1,0) and whose base curve
y = 0 ,x2 + y2 = 4.
ANS
The equation of any line through (1,1,0) is
(x -1)/ l = ( y-1 / m) = (z -0)/ n .--------(1)
It meets y = 0,where (x -1)/ l = (-1 / m) = (z -0)/ n .
Hence x = 1 –(l/m), z = -n/m.
Given that
x2 + y2 = 4 ---------(2)
y=0
--------- (3)
By (2)
(1 –(l/m))2 + (-n/m)2 = 4 ---And By (1),
l/m = (x-1)/(y-1)
n/m = z/(y-1).
Therefore by (4)
X2-3y2+z2-2xy+8y-4 =0,which is the required equation of cone.
13) Find the equation of the cone withn vertex at the origin whose guiding curve is
x 2 + y 2 + z 2 − x − 1 = 0, x 2 + y 2 + z 2 + y − 2 = 0
ANS
The guiding curve is a circle whose plane is
( x 2 + y 2 + z 2 + y − 2 )-( x 2 + y 2 + z 2 − x − 1 ) = 0 is x + y =1
∴ The equation of the cone is x 2 + y 2 + z 2 − x( x + y ) − ( x + y ) 2 = 0
⇒
x 2 + 3 xy − z 2 = 0
14) Find the equation of the sphere that passes through the circle
x 2 + y 2 + z 2 + 2 x − 2 y − 4 z − 22 = 0 , x + 2 y + 2 z + 7 = 0 and the point ( 1, -1, 2 )
ANS
The equation of the sphere is
( x 2 + y 2 + z 2 + 2 x − 2 y − 4 z − 22) + k ( x + 2 y + 2 z + 7) = 0 ----------(1 )
( 1 ) passes through ( 1, -1 ,2 )
∴ 6+2+2-8-22+k(1-2+4+7)=0
10k = 20
K=2
∴ The required sphere is x 2 + y 2 + z 2 + 4 x + 2 y − 8 = 0
PART - B
1)
Find the equation of the sphere that passes through the circle
x2+y2+z2-2x+3y-4z+6 = 0, 3x-4y+5z-15 = 0, and cuts the sphere
x2+y2+z2+2x+4y- 6z+11 = 0 orthogonally.
2) Find the tangent planes to the sphere x2+y2+z2-4x-2y-6z+5 = 0 which are parallel to
x+4y+8z=0. Find their points of contact.
3) Find the radius , centre and area of the circle in which the sphere
x2+y2+z2+2x-2y-4z-19=0 is cut by the plane x+2y+2z+7=0.
4) Find the two tangent plane to the sphere x2+y2+z2-4x+2y-6z+5=0, which are
parallel to the plane 2x+2y=z. Find their points of contact .
5) show that the plane 2x-2y+z=9 touches the
their points of contact.
sphere x2+y2+z2+2x+2y-7=0. And find
6) Prove that the plane 2x+2y-z=8 touches the
And find their points of contact.
sphere x2+y2+z2-4x+2y-6z+57=0.
7) .Find the equation of the tangent plane to the sphere 3 (x2+y2+z2)-2x-3y-4z-2=0 at
(1,2,3). Find also the equation of the normal to the sphere at ( 1,2,3).
8) Find the centre of the circle in which the sphere x2+y2+z2+2x-2y-4z-19=0 is cut by the
plane x+2y+2z+7=0
9) Verify whethere the spheres x2+y2+z2-2x+4y-6z+11 = 0 and x2+y2+z2+x-y+z-15 = 0
cut orthogonally.
10) Find the equation of the sphere which passes through the points (0,0,0),(0,1,-1),(-1,2,0)
and (1,2,3)
11) Find the equation of sphere having its centre on the plane 4x-5y-z = 3 and
Passing through the circle x2+y2+z2-2x-3y+4z+8 = 0; x-2y+z = 8
12) Find the equation of sphere that passes through the circle
x2+y2+z2+x-3y+2z-1=0;2x+5y-z+7=0 and cuts orthogonally the sphere
x2+y2+z2-3x+5y-7z-6=0.
13) If a line makes an angles α,β,γ,δ with the diagonals of a cube, then prove that
(i) Cos2α+ Cos2 β + Cos2 γ + Cos2 δ = 4/3.
(ii) Angle between any two diagonal is Cos-1(1/3)
14) Show that the equation of the right circular cylinder described on the circle through the
points (1,0,0), (0,1,0), (0,0,1) as the guiding curve is x2+y2+z2-zx-xy-yz-1=0.
15) Find the equation of the cone with vertex ( 5, 4, 3 ) and
3 x 2 + 2 y 2 = 6, y + z = 0 as base.
16)Find the equation of the cune whose vertex is (1 ,2, 3 ) and guiding curve is the circle
x 2 + y 2 + z 2 = 4, x + y + z = 1 .
17) Find the equation of the right circular cone generated by the straight lines drawn from the origin to cut
the circle through the three points (1,2,2), (2,1,-2) and (2,-2,1)
18) Find the equation of the right circular cylinder whose guiding circle is x 2 + y 2 + z 2 = 9, x − y + z = 3