80 12D Geometry

Show that
the
common Tangent the ellipse 1/r
of the cll:

angle 2 sin '(/,

1 / r - 1 esin O
sulbtend a n
2) al the for
and oints of
a r e two points of a cor
10. 1fSis
the focus,
t. such
locus ofthathe
equal lo 28. Prove that
constant and
rs) is is
at / and a Cor

intersection of
the langcnts
on the curve 1/r
(e focw
Il. f the normals at a , . y,ð :

+e cos 6:are co
then prove that:

tantan tantan =o
2

Also prove that if the four normals meet in p0int (p

an odd multiple of t.
=

+B+y +o 20
-

a
12. A circle passes through the point (,61) and touches the inin
from the pole, Show that its polar equation i
a distance c

2-2cr cos + -2c cose+

r Sin 6 nsin 6
of a conic, show that the polar o
13. Given the focus and directrix
conic passes through a fixed point.
point with respect to this
the circle which passes through the
14. Show that the equation to
1 e cos 6 at the point 6 a is =
touches the curve l/r = -

r(l- e coso
2 =
/ cos(0 -

o) -

el cos(6-2a)
ot
15. Ifthe normal at L, one of the extremities of the latus rectum
that
r = 1+ e cos 0 meets the curve again at Q, show

sO = -
(1+3e+)
(1+e-
 Saving ).12). i)
(2). 3). (4)
).(3). (4)
). (2). (4)
Thervefore thevertices of the tetrahedron are
o0 0,0) A(a.b,r) B(a
Let the equation of the required sphere be bc)and C(a b
y : 2ux +2vy+ 2wz
Since this sphere (5) passes through the above
d=0
point 6A
ab+c-2ua +2vb+ 2wc =0
a2+b2+c2 2ua-2vb +2wc =0
+
a2 +b+c +2ua [:d-0
+2vb-2wc 0 =

(8)(9) 2(a +b +c) +4ua =0

or, 2u -a
+b +c)Ja
=

(7)+(9) 2-(a +b2 +)/b

(7) (8)5 2w
-a +b2 +c*)/e
Substituting the values of u, v, w, d in (5), the required e

a+b +
Ex)2.A sphere of constantradius kpasses through the origi
the axesin A, B,
C; prove that the locus the centroid of ine
of
(a) triangle ABC is the sphere ty+-(4
(b) tetruhedron
0ABC is the sphere +y +
=(:"
Raj., B.Se., 2000,03; Ajmer,02,04; JounpurB.Se
Sol. By .6, the equation of the
sphere OABC S

a+y -ax by - C= 0
 - 2

the
vahes ol
t, , Y in (2), wen
shtain
Subetiing

ntreof the spthere

ofthe sphere OAR
the centr
radus rpasNes through the oelMenee
locus of
equinred

t h e a ris
whih the
es
tgphere f"locus of the footof the
4 ,B, C: P wt h a tt h elocus o ft h ef o a to f t " e a n e h n
nlar

perpende
ARs ghven by
from
Oe
the plane
Ajmer, B.Se., 03; Raj. R
the points A, Band Chbe
Sol. Let the co-ordinates ot
0) and (0, 0, c) respectisa
(a. 0, 0); (0, b,
the sphere
OABC is
cquation of
By S 1.6, the
2++ - ax - by - cz 0
=

is given to be r,
The radius ofthe sphere

( +b2 +c? 42
a =

Now equation of the plane ABC is

The equation ofthe line through O and perpendicular to the plane

A(say)
a /h
Eliminating a, b, c between (3) and (4), we get the locus ofthep

intersection of (3) and (4).

The co-ordinates ofany point on the line (4) are

Substituting the values of a, b, c in (2) and (3),

4 y 2+2)-4r
nd
(24y42=
 Sphfre 17)
A between (5) and (6),
Eliminatng
42
( :?)(r ?4y?+ 2)-
Hence Proved.
required locus
is the
which
equation of the
the splhere which passes throughthepoints
R.5. Find
0):(o, 1, 0) and(0,
0, 1) and has is radius as
smallas possible.
Raj. B.Sc., 03
Sol. Let the required
equation ofthe sphere be
2+y2
+ +2ux +2vy +2wz +d =
0 ..()
Since this sphere (1) passes through the points
(1,0, 0); (0, 1, 0) and (0, 0, 1),
1+2u+d=0 2u=-(d+1) ..(2)
1+2v+d =0 2v= -(d+1) ..3)
1+2w+d=0 2w=-(d+1) ...(4)
Ifthe radius of the sphere is r,then
=u+v2 +w2 -d
= 3(d+1d
4
[from (2), (3), (4)]

23d
+2d+ =f{d) (say)
4
Now for the minimum value of r, f(d) =0 ..(5)

Differentiating (5) wrt d, 6d+2 0

=
d=- ..(6)
Again for d= - , S"(d)>0
Therefore for d =- fld) i.e. i e . r will beminimum
Substituting this value of d in (2), (3) and (4),
2u= -2/3, 2v=-2/3, 2w -2/3 =

Substituting the values of u, v, w, d in (1)

+y +z2-(2/3)(x+y+ z)-=0
Ans.
Of, 3(x +y2+ z? -2 (x+y+z)-1=0
ExerciseI (a)
and radius is 2.
Find the equation of the sphere whose centre is (%,-, 1)
Find the centre and radius of the sphere
11.
r+y2 +z2 -2x+4y-6z
=

the points
ind equation of the sphere which passes through the
the
,-34); (1, -5, 2) and (1, -3, 0) and
whose centre lies on plane
ryreornt f

n (

) u(uhe)-(
b nhr'ra w'n
d

Second method: heequation

as 4byt+ 2ux +2vy47
be writen in the following form
CA

2w

Transferring the origin to the point

a b
ax +by2 +cr?=" a -d

This represents a cone if

.2Prove that theequation (fr)+ (y)+(h)=

a cone mat touches the co-ordinate planes and that the
eua
reciprocal cone is Jyz + gar + hxy = 0
[Ajmer B.Sc., 2000; Jodhgur
Sol. Given: x)+ (8y)+ (hz) = 0

Transposing and squaring, )+(y) =

Simplify and then again transposing and saquaring

Simplifying, fx* tg y +h22-2 ghyz -2hjr-

This is a homogeneous second
degree equation. Theree
a Cone whose vertex is at the
orngn.
Here noW
A
h0, ? - I ? 0, C-/
-1gh +fgh= 2fgh,(i f =2"
4 8 ) D(Geome
mutually J langentpl
)will have three plars f
Theconet ,
g e n e r a l o s , the e.conditon
the for w
mulunly

three
has (asn
one
(aah)(a's
a'W a ' )

Simphty m
locus is y
Hence the required
which is a
sphere
Exercise II (6)
Prove that
the cquat ion

2 22+2y24722-10)yz
10zx 2x +2y +27
conc wilh verlex (2, 2, 1).
representsa
between the lines
Prove that the angle
2
x+ytz=0 and ayz +bzx +Cxy =jis

(a) n/2,if a
th+c-0 Ajmer
(b) n/3,if'++' =0 Ajmer
a

3. If the plane ux + vy + wz = 0 cuts the cone ax + by4 + cz

pendicular lines, provethat

(b+c)u +(c+a) v2 +(a +b) w = 0

Raj. B.Sc.01H
4. Show that the lines drawn through the origin at right anglesto
planes of the cone ax +by2 + c2 = 0 generates the cone

a(b-c),b(c-a) c(a-b)-0
2
UdaipurB
5. Fnd the equation of thc tangent plane to the cone
42hxy 0 al the point
(a,p. 7). |Jodhpur B.Se.98,
6. Findthe angle bctween the
lines of intersection
with the cone 6
yz-27x +
ofucd
B.Sc., 02,
H
7. Find the equation 5xy 0 Raj.
to the lines in which the plane
cone 4x -y +3 0.
Ray. b

8. 1f the angle between the show

Jine planes through OX and (O1 D

of intersection lies on the cone

xy1=xy tan Udaipur

B.Sc. 01;
. Findthe locus of the ponts from Ajmer
to intersect the
which three mutuainy*
conic ax +by
). z =0
 Cone 53
ave threc ulually perpcnd
dicular Renerators, il
w A BC 0
he cotactors of a, b,
cofhe. .0)
4,B.
/ the
C,...espectively in the
deteminant
|hb

Ab . B=ac-g, C= ab-h2
hstitutingthe values of A, B, Cin (1),
be+ ca + ab = f2 +g2 + h2
the required condition.
Hence Proved.
Exercise II (¢)
1. Find the equation of a right circul cone whose vertex, axis and semi
are as follows
erticalangle
)2.1): =y-2=

)(0.0.0)- Udaipur B.Sc. 02]

() (1.1.1);
0 , 0 , 0); axis OZ
al.aj. B.Sc.,98, Hons., 991
Find the equation of a right circular cone whose
vertex and
areas follows guiding curve
(a) (1.2,3), + y +z2 =4. z 0 =

(b) (0.0.0) +y +z2

=a*, =a/2
.Find the equation to the
right circular cone having vertex at the
=y= and 2x
3y=
5: as one of the
= -
origin, axis
generators.
1a
right circular JAjmer B.Sc. 04; Hons. 01
find its
cone has three mutually perpendicular tangent planes,
semi-vertical angle.
ICIeSenis
Cof2
onetheof tee
neratiators of the one mutuafly pependisulr
oher two.
JodlipurB.>.
 7he vtei ot he coe i fn h.) and
AY7 plane
cuta t
: 0 0 Show that the
7-plane cuts m
t
hs ha
0. 0 |UdaipurB.Se., 01
Pove thal the ines from the origin se Ray.R
7 to touch the
4: 2u1 2vy + 2wz td - 0 lie on the.
COne
a +:= (ux+ vy +wz
S Show that the general cquation to a cone which touc

planesis the ctr

a'+b2 -

2bcyz -2cazx 2 abxy:

xy = 0
(ax) (bv) + (er) =
0

Ajmer B.Sc. 01;Jodhpur 04; Raj. B.Sc.

9. Show that the plane ax + by + cz = 0 cuts the cone yz+ zx Hons
lines inclined at an angle:

tan
Ja2 +b2 +c2)(a +b +c-2bc -2ca-2a
bc +ca+ab
n
Prove that the tangent planes to the cone
fyz+gr hy
perpendicular to the generators of the cone

22+g2y2 +h-2ghyz-2lifex-2fsw =0
Exercise II (d)
Multiple ChoiceQuestions (MCQ): (Q. 1-9)
. Ifa right circular cone has three mutually perpendicular generatin
ils semivertical angle is:

(A) lan (B) lan RAS (Pre

(C) (D) 1an2

. t h ecquation of a cone is Nby+ 0 . then the

RAS(Pr
TeCprocalcone s:
 on the line
irne throwgh
be anypoint
{o
R0}
this pomt
satisfies(2) alse
this line representssthe
M
on his
PI
hen

There
fore
every
point rtare tep
by () is the lo

Thus the
srface
represented
the Vnen
7-4N15 a cylinder
whose geaeraator
s
Hence (1) represents
f(2, x) = 0
parallel e
f(y.)=0,.
Similarly

cylinder
whose generators
are parallel to x ar y-27is te
represents
a

llustrative Examples
E. 1. Find the cquation of
the cylinder whosegeneres
and whose guiding
nthe line
x/l =
y/-2 =z/3 curve i fhe
cue

2 y = 1 , z =0. Raj. B.Sc. 03; Hons

Sol. Let (a, B. Y) be any point on the Theaen
cylinder, The

this point is
y- -7
generator through
the plane =
0 the
Clearly this generator meets at
z noi

a-B+
Ifthis point lies on the curve +2 =1 , then

-+2B
9u+97+18 -6a7+12ß7=9
Therefore the locus of (a, B. Y) is

9+2+z2)-6x:+12yT =9
which is the required equation of the cylinder.
EAind the equution of a right circular crlinder hR
X - -2 and which pusses through (0, D,
2 1).|
Raj. B.Sc.01:H
S91. From the figure. the radius of the cylinde? 1
distaiie of the
point A(0,0.11 to the axis of he cy i1
Therefore
AA=AB- BA
 ylnle 63)
o-2+ (0 1(1o']
[(0-2.2 +(0- 1).1 4+(1-0).3
4+1+9) 0,0,1)
M
4 40

614 G I( 2,1.0)
be any point the
LetP(1, y, 3) on
cylinder, then
Ney
PB BO -
= Op? Axis
Fig.3.3
alle a-2+(-if +(:-0}]
pse
04 [(r-2).2+(y-1).1+(z-0).3
V(4+1+9)
B(2,1.0)

Or 10x2+13y2 +5z' -6yz -12zx (x,y.2)P

-4xy -36x 18y +30z -35-

0
which is the required equation of the
Axis
riehtcircular cylinder. Fig.3.4
(Ex.3Find the equation of a right circular cylinder whose radius
ura uxis is r =2y =
-z. Prove that the area is 4
theplane XO Yis 24 n. of the section of this cylinder by
Raj. B.Sc. 99; Hons. 0i
Sol. Let P(x, y, z) be any
point on the cylinder as shown in the figure.
Therefore AP?-OA2 =OP2
0
(r-0)+(y-0) +(2-1
(r-0).2+(y - 0).1+(3-0-2)1 A(0,0,0)
(4+1+4)
5%+82+572+4
which is the ZX-4xy-144=0 .. (1)
ne
required equation of the cylinder.
and equation of the plane through the origin
perpendicular to the asis of the cylinder is

ZX+y-2: = 0 ... (2) ANIS

Thearea of the section ofthis
cylinderby the Fig.3.5
 De
Therefore loc. fP(a. B. y) is Cylinde ( 65
L4y42-av-by-ca)=(++2
is
the required equation of the cylinder. Ans.
3 Axis of the cylinder
hat the axis of the
ed We
We
know
cylinder passes through the
ling 2. ) of the circle and its DC's are
proportional to 1/a, 1/b,centre
(al2. b/2, e / 2 )

T h e r e f o r e .
the requir equation of the axis is
1/c.
03 x-ay-;b3
1/a
) d the equation o a right crcular Ans.
E|(Ex.5toaxis and
intersect the cylinder whose
a r

allel
ep a r a l e

surfaces ax generators
p. tIz TUdaipur B.Sc. 2000; +by2 + cz2 =1 and
=

r +1y
ol. If P(a, B. Y) is any point on the
Ajmer B.Sc. 99, 041
generator, then its equation will
he
- y-5 -Y
0 0 (say)
are fore the co-ordinates of any
point on the generator are(a, B,
rhic noint will lie on
ax y+ r)
+
by- + czá 1, lx+my
=

+nz p =
if aa +bB +c(y+r) = 1
and (1)
fa+mp+n(y+r)= p
Eliminatingrfrom(1 )and (2), 2)

aa +bß-+c| YP-fa+ B+ny)]= 1

Therefore locus of P (a, B. y) is

ax+by2 + CP=t - my)*-=1

an+c) x - (bn +cm>) y +2cimxy

nch is the -2cpix-2cpmy + cp - n = 0
required quation of the
EA. 6.
Find the cylinder.
equation of the cylinder whose
he line ri
generating lines are
t- y m =zin and which touches the
=a.
=

spliere
Sol. f Raj. B.Se. 03: Udaipur B.Sc.. 01; Ajmer
B.Sc.,04
P S. is anN point on the to cylinder. then any line parallel
tm=:
 Cylinder 677,
ofthe circle (9-1)= V8
=

radius

herefore plane will be (1, -2, 2).

to the O (0, 0, 0)
normal
norn
line with DR's
h e
the line
equations
f the
of
o f

he
R's
herefore
h erefore
the origin O(0,0,0)
through
o) and
n

Fig3.9
on the cylinder
be any point
/1. y ,
Plr,
V. 2)
Let axis.

the

(000)
on PB2= PA2-AB2 AJ(0,0,0)
pd Therefore

s-r-0+(-0+(-o]
8
1B
2 } P(x, y, z)
-0).2
r-0.1+(-0).(-2)+
(
9
Sty+-1/9(r
-2jy+2-)2
72 0 Fig3.10
4zx -
=

+4xy +8yz
-

r+5y +5z"
the cylinder.
require
equation of
the
ichis
Exercise III(a)
are as
radius and axis
circular cylinder whose
Find the equation
ofright
follows [Raj. B.Sc., 04]
y-2 =
0) 2 [Raj. B.Sc., 99]
(b) 3, [Raj. B.Sc., 03]

) 2 2 [Raj. B. Sc., 01]

4 [Raj. B. Sc., 02]

is a
2 whose radius
circular cylinder
Find th e cartesian equation of a right
andaxis is 2-axis. guiding
circlepasses

cylinderwhose of
ndthe equation ofthe rightcircular
Find
find the equation
0, 1).Also
(0,
through the points (1,0,0), (0, 1,0)andB.Sc., 04; A4jmer,
01; Hons. 03
ls axis. Udaipur
 circulatc
4 nd the cqualion
of the right
ylinder whoe gnding
9,
crcular cylinder
aj. B..,01, n
of.

Findthe cquat0n otaight

thruph (d,2. 3 )and ' s are proportional 2,, 3,6.

of cylinder whose generafco

rators are
the cquation
6. Find
and whose guiding curve is the ellipse ,2
parallel to the
+

Raj. B.Sc, 02;

Find the cquation
to the cylinder whose generators
ors are Hom
and intersects the curve ar" +by2 = 2z, lx + my +nz parallej t6 2
p.
IHint:Eliminate z between ax +by =2z and
s. Find the equation of a cylinder whose generators are+my+
nz= n.=
arallel
z/n and base is the curve x2 + y2 +2 arallel,
x/l y/m=
=

=
RAS 91; Ajmer, B.Se
Find the equation generator
of the cylinder whose
rators are paralle
9.
x/4 = y/-2= z/3 and whichintersects the ellipse 4 2 3.Pr =1, t
Raj. B.S
10. Find the equation ofthe quadric cylinder having generators parall
axis and intersecting the curve ax +by + cz2 =1, lx+my-
11. Find the equation ofa right circular cylinder whose radius is? am
and is parallel to x-axis.
axis passes through the point (1, 2, 3)
Raj. B.Sc., 02; Ajmer,
circular cylinder whose radius is 3 and wt
12. Find the equation of a right
axis passes through the point (1, -1,2) having direction
ratios 2-:
Raj.B.Sc,
cylinder which envelopes the conic
13. Find the equation of a

x2/a2+y2/b2 +z2/c2=1 and whosegeneratorsareparalleltothe

having DC's l, m, n.
at any point (a, B. Y)0
14. Show that the equation of the tangent plane
0 is.
cylinder ax2 +2hxy + by2 +2gx +2fy+c = fß+c=0
x{0a + hB+8)+ y(ha+ bß+ f)+ ga+ g e n e r a t o r t h r o

ofthe geneta
Also show that it touches the cylinder at all points
that point. x2+y2+72-2x+4
15. Find the enveloping ofcylinder ofthe sphere Also
finditsgud

the line. x y=z.AID

2. =
Hons
having its generators parallel to B . S c . 01;
IRaj.
whose
Curve. cylinder through the
the f
16. Find the equation of the right circular oeh
which passes
(x-2)/1 =(y-1)/1 0)/3 and