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Edurev123
7. Ellipsoid and its Properties
7.1 Three points ?? ,?? ,?? are taken on the ellipsoid
?? ?? ?? ?? +
?? ?? ?? ?? +
?? ?? ?? ?? =?? so that the lines
joining ?? ,?? ,?? to the origin are mutually perpendicular: Prove that the plane ??????
touches a fixed sphere.
(2011 : 20 Marks)
Solution:
Let the equation of the plane ?????? be
???? +???? +???? =1 (??)
The equation of the given ellipsoid is
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=1 (???? )
The equation of the cone with vertex at (0,0,0) and the curve of intersection of (i) and the
ellipsoid (ii) as the guiding curve is
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=(???? +???? +???? )
2
(?????? )
If the cone (iii) has three mutually perpendicular generators then
Coefficient of ?? 2
+ Coefficient of ?? 2
+ Coefficient of ?? 2
=0
? (?? 2
-
1
?? 2
)+(?? 2
-
1
?? 2
)+(?? 2
-
1
?? 2
)=0
? ?? 2
+?? 2
+?? 2
=
1
?? 2
+
1
?? 2
+
1
?? 2
=
1
?? 2
(?????? ) (???? )
If the plane (i) touches the sphere ?? 2
+?? 2
+?? 2
=?? 2
, then the length of the perpendicular
from the centre (0,0,0) of the sphere to (i) must be equal to the radius ?? of the sphere.
i.e.,
1
v?? 2
+?? 2
+?? 2
=??
? ?? 2
+?? 2
+?? 2
=
1
?? 2
, which is true by virtue of (iv).
Hence, the plane (i) touches the sphere ?? 2
+?? 2
+?? 2
=?? 2
.
Page 2
Edurev123
7. Ellipsoid and its Properties
7.1 Three points ?? ,?? ,?? are taken on the ellipsoid
?? ?? ?? ?? +
?? ?? ?? ?? +
?? ?? ?? ?? =?? so that the lines
joining ?? ,?? ,?? to the origin are mutually perpendicular: Prove that the plane ??????
touches a fixed sphere.
(2011 : 20 Marks)
Solution:
Let the equation of the plane ?????? be
???? +???? +???? =1 (??)
The equation of the given ellipsoid is
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=1 (???? )
The equation of the cone with vertex at (0,0,0) and the curve of intersection of (i) and the
ellipsoid (ii) as the guiding curve is
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=(???? +???? +???? )
2
(?????? )
If the cone (iii) has three mutually perpendicular generators then
Coefficient of ?? 2
+ Coefficient of ?? 2
+ Coefficient of ?? 2
=0
? (?? 2
-
1
?? 2
)+(?? 2
-
1
?? 2
)+(?? 2
-
1
?? 2
)=0
? ?? 2
+?? 2
+?? 2
=
1
?? 2
+
1
?? 2
+
1
?? 2
=
1
?? 2
(?????? ) (???? )
If the plane (i) touches the sphere ?? 2
+?? 2
+?? 2
=?? 2
, then the length of the perpendicular
from the centre (0,0,0) of the sphere to (i) must be equal to the radius ?? of the sphere.
i.e.,
1
v?? 2
+?? 2
+?? 2
=??
? ?? 2
+?? 2
+?? 2
=
1
?? 2
, which is true by virtue of (iv).
Hence, the plane (i) touches the sphere ?? 2
+?? 2
+?? 2
=?? 2
.
7.2 Find the length of the normal chord through a point ?? of the ellipsoid
?? ?? ?? ?? +
?? ?? ?? ?? +
?? ?? ?? ?? =??
and prove that if it is equal to ?? ?? ?? ?? , where ?? ?? is the point where the nomal chord
through ?? meets the ???? -plane, then ?? lies on the cone
?? ?? ?? ?? (?? ?? ?? -?? ?? )+
?? ?? ?? ?? (?? ?? ?? -?? ?? )+
?? ?? ?? ?? =??
(2019 : 15 Marks)
Solution:
Let ?? be (?? ,?? ,?? ) , then the equations of the normal to the given ellipsoid at ?? (?? ,?? ,?? ) are
?? -?? (???? /?? 2
)
=
?? -?? (???? /?? 2
)
=
?? -?? (???? /?? 2
)
=?? (say) (1)
1
?? 2
=
?? 2
?? 4
+
?? 2
?? 4
+
?? 2
?? 4
(2)
? The co-ordinates of any point ?? on the normal (1) are (?? +
????
?? 2
,?? +
????
?? 2
?? ,?? +
????
?? 2
?? )
where ?? is the distance of ?? from ?? .
If ?? lies on the given ellipsoid i.e., ???? is the normal chord, then
1
?? 2
(?? +
????
?? 2
?? )
2
+
1
?? 2
(?? +
????
?? 2
?? )
2
+
1
?? 2
(?? +
????
?? 2
)
2
=1
=?? 2
?? 2
(
?? 2
?? 6
+
?? 2
?? 6
+
?? 2
?? 6
)+2???? (
?? 2
?? 4
+
?? 2
?? 4
+
?? 4
?? 4
)+(
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
)=1
=?? 2
?? 2
(
?? 2
?? 6
+
?? 2
?? 6
+
?? 2
?? 6
)+2???? (
1
?? 2
)=0
From (2) and S
?? 2
?? 2
=1 as ?? (?? ,?? ,?? ) lies on the given coincoid.
?? =
-2
?? 3
(
?? 2
?? 6
+
?? 2
?? 6
+
?? 2
?? 6
)
= length of normal chord ???? (3)
Also, let the normal at ?? (?? ,?? ,?? ) meets the coordinate planes viz., ???? ,???? and ???? planes
at ?? 1
,?? 2
and ?? 3
then puling ?? =0,?? =0 and ?? =0 in succession in the eqn. (1), we have
respectively,
Given,
?? ?? 1
=-
?? 2
?? ,?? ?? 2
=-
?? 2
?? and ?? ?? 3
=
?? 2
?? (4)
Page 3
Edurev123
7. Ellipsoid and its Properties
7.1 Three points ?? ,?? ,?? are taken on the ellipsoid
?? ?? ?? ?? +
?? ?? ?? ?? +
?? ?? ?? ?? =?? so that the lines
joining ?? ,?? ,?? to the origin are mutually perpendicular: Prove that the plane ??????
touches a fixed sphere.
(2011 : 20 Marks)
Solution:
Let the equation of the plane ?????? be
???? +???? +???? =1 (??)
The equation of the given ellipsoid is
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=1 (???? )
The equation of the cone with vertex at (0,0,0) and the curve of intersection of (i) and the
ellipsoid (ii) as the guiding curve is
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=(???? +???? +???? )
2
(?????? )
If the cone (iii) has three mutually perpendicular generators then
Coefficient of ?? 2
+ Coefficient of ?? 2
+ Coefficient of ?? 2
=0
? (?? 2
-
1
?? 2
)+(?? 2
-
1
?? 2
)+(?? 2
-
1
?? 2
)=0
? ?? 2
+?? 2
+?? 2
=
1
?? 2
+
1
?? 2
+
1
?? 2
=
1
?? 2
(?????? ) (???? )
If the plane (i) touches the sphere ?? 2
+?? 2
+?? 2
=?? 2
, then the length of the perpendicular
from the centre (0,0,0) of the sphere to (i) must be equal to the radius ?? of the sphere.
i.e.,
1
v?? 2
+?? 2
+?? 2
=??
? ?? 2
+?? 2
+?? 2
=
1
?? 2
, which is true by virtue of (iv).
Hence, the plane (i) touches the sphere ?? 2
+?? 2
+?? 2
=?? 2
.
7.2 Find the length of the normal chord through a point ?? of the ellipsoid
?? ?? ?? ?? +
?? ?? ?? ?? +
?? ?? ?? ?? =??
and prove that if it is equal to ?? ?? ?? ?? , where ?? ?? is the point where the nomal chord
through ?? meets the ???? -plane, then ?? lies on the cone
?? ?? ?? ?? (?? ?? ?? -?? ?? )+
?? ?? ?? ?? (?? ?? ?? -?? ?? )+
?? ?? ?? ?? =??
(2019 : 15 Marks)
Solution:
Let ?? be (?? ,?? ,?? ) , then the equations of the normal to the given ellipsoid at ?? (?? ,?? ,?? ) are
?? -?? (???? /?? 2
)
=
?? -?? (???? /?? 2
)
=
?? -?? (???? /?? 2
)
=?? (say) (1)
1
?? 2
=
?? 2
?? 4
+
?? 2
?? 4
+
?? 2
?? 4
(2)
? The co-ordinates of any point ?? on the normal (1) are (?? +
????
?? 2
,?? +
????
?? 2
?? ,?? +
????
?? 2
?? )
where ?? is the distance of ?? from ?? .
If ?? lies on the given ellipsoid i.e., ???? is the normal chord, then
1
?? 2
(?? +
????
?? 2
?? )
2
+
1
?? 2
(?? +
????
?? 2
?? )
2
+
1
?? 2
(?? +
????
?? 2
)
2
=1
=?? 2
?? 2
(
?? 2
?? 6
+
?? 2
?? 6
+
?? 2
?? 6
)+2???? (
?? 2
?? 4
+
?? 2
?? 4
+
?? 4
?? 4
)+(
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
)=1
=?? 2
?? 2
(
?? 2
?? 6
+
?? 2
?? 6
+
?? 2
?? 6
)+2???? (
1
?? 2
)=0
From (2) and S
?? 2
?? 2
=1 as ?? (?? ,?? ,?? ) lies on the given coincoid.
?? =
-2
?? 3
(
?? 2
?? 6
+
?? 2
?? 6
+
?? 2
?? 6
)
= length of normal chord ???? (3)
Also, let the normal at ?? (?? ,?? ,?? ) meets the coordinate planes viz., ???? ,???? and ???? planes
at ?? 1
,?? 2
and ?? 3
then puling ?? =0,?? =0 and ?? =0 in succession in the eqn. (1), we have
respectively,
Given,
?? ?? 1
=-
?? 2
?? ,?? ?? 2
=-
?? 2
?? and ?? ?? 3
=
?? 2
?? (4)
?????????? , ???? =4?? ?? 3
???? =4(-
?? 2
?? )
?
-2
?? 3
(
?? 2
?? 6
+
?? 2
?? 6
+
?? 2
?? 6
)
=4(-
?? 2
?? )
?
?? 2
?? 6
(2?? 2
-?? 2
)+
?? 2
?? 6
(2?? 2
-?? 2
)+
?? 2
?? 6
(2?? 2
-?? 2
)=0
? The locus of ?? (?? ,?? ,?? ) is
?? 2
?? 5
(2?? 2
-?? 2
)+
?? 2
?? 6
(2?? 2
-?? 2
)+
?? 2
?? 4
=0. Hence, Proved.
7.3 Find the equations of the tangent plane to the ellipsoid ?? ?? ?? +?? ?? ?? +?? ?? ?? =????
which passes through the line ?? -?? -?? =?? =?? -?? +?? ?? -?? .
(2020 : 10 Marks)
Solution:
Given line ?? -?? -?? =0-?? -?? +2?? -9 (??)
Equation of any plane through given line (i)
?? -?? -?? +?? (?? -?? +2?? -9)=0
? ?? (1+?? )+?? (-1-?? )+?? (-1+2?? )=9?? (???? )
If this plane (ii) touches given ellipsoid then applying
?? 2
?? +
?? 2
?? +
?? 2
?? =?? 2
(?????? )
Given ellipsoid
2?? 2
+6?? 2
+3?? 2
=27
?
2
27
?? 2
+
6
27
?? 2
+
3
27
?? 2
=1
?
2
27
?? 2
+
2
9
?? 2
+
1
9
?? 2
=1
? ?? =
2
27
,?? =
2
9
,?? =
1
9
So from (iii)
Page 4
Edurev123
7. Ellipsoid and its Properties
7.1 Three points ?? ,?? ,?? are taken on the ellipsoid
?? ?? ?? ?? +
?? ?? ?? ?? +
?? ?? ?? ?? =?? so that the lines
joining ?? ,?? ,?? to the origin are mutually perpendicular: Prove that the plane ??????
touches a fixed sphere.
(2011 : 20 Marks)
Solution:
Let the equation of the plane ?????? be
???? +???? +???? =1 (??)
The equation of the given ellipsoid is
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=1 (???? )
The equation of the cone with vertex at (0,0,0) and the curve of intersection of (i) and the
ellipsoid (ii) as the guiding curve is
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
=(???? +???? +???? )
2
(?????? )
If the cone (iii) has three mutually perpendicular generators then
Coefficient of ?? 2
+ Coefficient of ?? 2
+ Coefficient of ?? 2
=0
? (?? 2
-
1
?? 2
)+(?? 2
-
1
?? 2
)+(?? 2
-
1
?? 2
)=0
? ?? 2
+?? 2
+?? 2
=
1
?? 2
+
1
?? 2
+
1
?? 2
=
1
?? 2
(?????? ) (???? )
If the plane (i) touches the sphere ?? 2
+?? 2
+?? 2
=?? 2
, then the length of the perpendicular
from the centre (0,0,0) of the sphere to (i) must be equal to the radius ?? of the sphere.
i.e.,
1
v?? 2
+?? 2
+?? 2
=??
? ?? 2
+?? 2
+?? 2
=
1
?? 2
, which is true by virtue of (iv).
Hence, the plane (i) touches the sphere ?? 2
+?? 2
+?? 2
=?? 2
.
7.2 Find the length of the normal chord through a point ?? of the ellipsoid
?? ?? ?? ?? +
?? ?? ?? ?? +
?? ?? ?? ?? =??
and prove that if it is equal to ?? ?? ?? ?? , where ?? ?? is the point where the nomal chord
through ?? meets the ???? -plane, then ?? lies on the cone
?? ?? ?? ?? (?? ?? ?? -?? ?? )+
?? ?? ?? ?? (?? ?? ?? -?? ?? )+
?? ?? ?? ?? =??
(2019 : 15 Marks)
Solution:
Let ?? be (?? ,?? ,?? ) , then the equations of the normal to the given ellipsoid at ?? (?? ,?? ,?? ) are
?? -?? (???? /?? 2
)
=
?? -?? (???? /?? 2
)
=
?? -?? (???? /?? 2
)
=?? (say) (1)
1
?? 2
=
?? 2
?? 4
+
?? 2
?? 4
+
?? 2
?? 4
(2)
? The co-ordinates of any point ?? on the normal (1) are (?? +
????
?? 2
,?? +
????
?? 2
?? ,?? +
????
?? 2
?? )
where ?? is the distance of ?? from ?? .
If ?? lies on the given ellipsoid i.e., ???? is the normal chord, then
1
?? 2
(?? +
????
?? 2
?? )
2
+
1
?? 2
(?? +
????
?? 2
?? )
2
+
1
?? 2
(?? +
????
?? 2
)
2
=1
=?? 2
?? 2
(
?? 2
?? 6
+
?? 2
?? 6
+
?? 2
?? 6
)+2???? (
?? 2
?? 4
+
?? 2
?? 4
+
?? 4
?? 4
)+(
?? 2
?? 2
+
?? 2
?? 2
+
?? 2
?? 2
)=1
=?? 2
?? 2
(
?? 2
?? 6
+
?? 2
?? 6
+
?? 2
?? 6
)+2???? (
1
?? 2
)=0
From (2) and S
?? 2
?? 2
=1 as ?? (?? ,?? ,?? ) lies on the given coincoid.
?? =
-2
?? 3
(
?? 2
?? 6
+
?? 2
?? 6
+
?? 2
?? 6
)
= length of normal chord ???? (3)
Also, let the normal at ?? (?? ,?? ,?? ) meets the coordinate planes viz., ???? ,???? and ???? planes
at ?? 1
,?? 2
and ?? 3
then puling ?? =0,?? =0 and ?? =0 in succession in the eqn. (1), we have
respectively,
Given,
?? ?? 1
=-
?? 2
?? ,?? ?? 2
=-
?? 2
?? and ?? ?? 3
=
?? 2
?? (4)
?????????? , ???? =4?? ?? 3
???? =4(-
?? 2
?? )
?
-2
?? 3
(
?? 2
?? 6
+
?? 2
?? 6
+
?? 2
?? 6
)
=4(-
?? 2
?? )
?
?? 2
?? 6
(2?? 2
-?? 2
)+
?? 2
?? 6
(2?? 2
-?? 2
)+
?? 2
?? 6
(2?? 2
-?? 2
)=0
? The locus of ?? (?? ,?? ,?? ) is
?? 2
?? 5
(2?? 2
-?? 2
)+
?? 2
?? 6
(2?? 2
-?? 2
)+
?? 2
?? 4
=0. Hence, Proved.
7.3 Find the equations of the tangent plane to the ellipsoid ?? ?? ?? +?? ?? ?? +?? ?? ?? =????
which passes through the line ?? -?? -?? =?? =?? -?? +?? ?? -?? .
(2020 : 10 Marks)
Solution:
Given line ?? -?? -?? =0-?? -?? +2?? -9 (??)
Equation of any plane through given line (i)
?? -?? -?? +?? (?? -?? +2?? -9)=0
? ?? (1+?? )+?? (-1-?? )+?? (-1+2?? )=9?? (???? )
If this plane (ii) touches given ellipsoid then applying
?? 2
?? +
?? 2
?? +
?? 2
?? =?? 2
(?????? )
Given ellipsoid
2?? 2
+6?? 2
+3?? 2
=27
?
2
27
?? 2
+
6
27
?? 2
+
3
27
?? 2
=1
?
2
27
?? 2
+
2
9
?? 2
+
1
9
?? 2
=1
? ?? =
2
27
,?? =
2
9
,?? =
1
9
So from (iii)
27
2
(1+?? )
2
+
9
2
(-1-?? )
2
+9(2?? -1)
2
=(9?? )
2
? 27(1+?? )
2
+9(-1-?? )
2
+18(2?? -1)
2
=162?? 2
? ?? 2
[27+9+72-162]+2?? [27+9-36]+27+9+18=0
? -54?? 2
+54=0
? ?? 2
=1
? ?? =±1
Required equations of tangent planes will be:
if ?? =1; (ii) =2?? -2?? +?? =9
if ?? =-1; (ii) =?? =3.
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Nov 21, 2024 Last updated |
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