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Page 1
Edurev123
5. Cone and its properties
5.1 Prove that the normal from the point (?? ,?? ,?? ) to the paraboloid
?? ?? ?? ?? +
?? ?? ?? ?? =?? ?? be
on the cone
?? ?? -?? -
?? ?? -?? +
?? ?? -?? ?? ?? -?? =??
(Note: There is an error in the question of (+) sign instead of (-) before second
term.)
(2009 : 20 Marks)
Solution:
Approach : From the general equation of normal passing through a point (?? ,?? ,?? )
eliminate the direction cosines.
?? 2
?? 2
+
?? 2
?? 2
=2?? (??)
is given equation of paraboloid equation of tangent plane to paraboloid at (?? ,?? ,h) is
????
?? 2
+
????
?? 2
=(?? +h)
? Normal to paraboloid at (?? ,?? ,h) has direction cosines (
?? ?? 2
,
?? ?? 2
,-1) and the equation of
normal is
?? 2
(?? -?? )
?? =
?? 2
(?? -?? )
?? =
?? -h
-1
It passes through a point (?? ,?? ,?? ) if
?? 2
(?? -?? )
?? =
?? 2
(?? -?? )
?? =
?? -h
-1
=?? (let)
? ?? =
?? 2
?? ?? 2
+?? ;?? =
?? 2
?? ?? 2
+?? ;h=?? +??
Now let any normal through (?? ,?? ,?? ) be
Page 2
Edurev123
5. Cone and its properties
5.1 Prove that the normal from the point (?? ,?? ,?? ) to the paraboloid
?? ?? ?? ?? +
?? ?? ?? ?? =?? ?? be
on the cone
?? ?? -?? -
?? ?? -?? +
?? ?? -?? ?? ?? -?? =??
(Note: There is an error in the question of (+) sign instead of (-) before second
term.)
(2009 : 20 Marks)
Solution:
Approach : From the general equation of normal passing through a point (?? ,?? ,?? )
eliminate the direction cosines.
?? 2
?? 2
+
?? 2
?? 2
=2?? (??)
is given equation of paraboloid equation of tangent plane to paraboloid at (?? ,?? ,h) is
????
?? 2
+
????
?? 2
=(?? +h)
? Normal to paraboloid at (?? ,?? ,h) has direction cosines (
?? ?? 2
,
?? ?? 2
,-1) and the equation of
normal is
?? 2
(?? -?? )
?? =
?? 2
(?? -?? )
?? =
?? -h
-1
It passes through a point (?? ,?? ,?? ) if
?? 2
(?? -?? )
?? =
?? 2
(?? -?? )
?? =
?? -h
-1
=?? (let)
? ?? =
?? 2
?? ?? 2
+?? ;?? =
?? 2
?? ?? 2
+?? ;h=?? +??
Now let any normal through (?? ,?? ,?? ) be
?? -?? ?? =
?? -?? ?? =
?? -?? ?? (???? )
Then,
?? ?? /?? 2
=
?? ?? /?? 2
=
?? -1
(any normal must have such d.c.'s)
?
??(?? 2
+?? )
?? =
?? (?? 2
+?? )
?? =
?? -1
?
?? -1
=
?? 2
-?? 2
?? ?? -
?? ?? ? ?? ?? (
?? ?? -
?? ?? )=?? 2
-?? 2
? Replacing ??,?? ,?? from (ii)
?? ?? -?? -
?? ?? -?? +
?? 2
-?? 2
?? -?? =0
5.2 Show that the cone ???? +???? +???? =?? cuts the sphere ?? ?? +?? ?? +?? ?? =?? ?? in two
equal circles, and find their area.
(2011 : 20 Marks)
Solution:
The given equations are
?? 2
+?? 2
+?? 2
=?? 2
(??)
?????? ???? +???? +???? =0 (???? )
Multiply (ii) by 2 and add it to (i), we get
or
?? 2
+?? 2
+?? 2
+2(???? +2?? +???? ) =?? 2
(?? +?? +?? )
2
=?? 2
??? +?? +?? =±??
? The equations of the required circles are ·
?? 2
+?? 2
+?? 2
=?? 2
,?? +?? +?? =?? (?????? )
?? 2
+?? 2
+?? 2
=?? 2
,?? +?? +?? =-?? (???? )
Area of Circle (i)
Centre of the sphere ?? 2
+?? 2
+?? 2
=?? 2
is (0,0,0) .
If ?? 1
is the length of perpendicular from the centre (0,0,0) of the sphere ?? 2
+?? 2
+?? 2
=?? 2
to the plane ?? +?? +?? =?? , then
?? 1
=|
0+0+0-?? v1+1+1
|=
?? v3
Page 3
Edurev123
5. Cone and its properties
5.1 Prove that the normal from the point (?? ,?? ,?? ) to the paraboloid
?? ?? ?? ?? +
?? ?? ?? ?? =?? ?? be
on the cone
?? ?? -?? -
?? ?? -?? +
?? ?? -?? ?? ?? -?? =??
(Note: There is an error in the question of (+) sign instead of (-) before second
term.)
(2009 : 20 Marks)
Solution:
Approach : From the general equation of normal passing through a point (?? ,?? ,?? )
eliminate the direction cosines.
?? 2
?? 2
+
?? 2
?? 2
=2?? (??)
is given equation of paraboloid equation of tangent plane to paraboloid at (?? ,?? ,h) is
????
?? 2
+
????
?? 2
=(?? +h)
? Normal to paraboloid at (?? ,?? ,h) has direction cosines (
?? ?? 2
,
?? ?? 2
,-1) and the equation of
normal is
?? 2
(?? -?? )
?? =
?? 2
(?? -?? )
?? =
?? -h
-1
It passes through a point (?? ,?? ,?? ) if
?? 2
(?? -?? )
?? =
?? 2
(?? -?? )
?? =
?? -h
-1
=?? (let)
? ?? =
?? 2
?? ?? 2
+?? ;?? =
?? 2
?? ?? 2
+?? ;h=?? +??
Now let any normal through (?? ,?? ,?? ) be
?? -?? ?? =
?? -?? ?? =
?? -?? ?? (???? )
Then,
?? ?? /?? 2
=
?? ?? /?? 2
=
?? -1
(any normal must have such d.c.'s)
?
??(?? 2
+?? )
?? =
?? (?? 2
+?? )
?? =
?? -1
?
?? -1
=
?? 2
-?? 2
?? ?? -
?? ?? ? ?? ?? (
?? ?? -
?? ?? )=?? 2
-?? 2
? Replacing ??,?? ,?? from (ii)
?? ?? -?? -
?? ?? -?? +
?? 2
-?? 2
?? -?? =0
5.2 Show that the cone ???? +???? +???? =?? cuts the sphere ?? ?? +?? ?? +?? ?? =?? ?? in two
equal circles, and find their area.
(2011 : 20 Marks)
Solution:
The given equations are
?? 2
+?? 2
+?? 2
=?? 2
(??)
?????? ???? +???? +???? =0 (???? )
Multiply (ii) by 2 and add it to (i), we get
or
?? 2
+?? 2
+?? 2
+2(???? +2?? +???? ) =?? 2
(?? +?? +?? )
2
=?? 2
??? +?? +?? =±??
? The equations of the required circles are ·
?? 2
+?? 2
+?? 2
=?? 2
,?? +?? +?? =?? (?????? )
?? 2
+?? 2
+?? 2
=?? 2
,?? +?? +?? =-?? (???? )
Area of Circle (i)
Centre of the sphere ?? 2
+?? 2
+?? 2
=?? 2
is (0,0,0) .
If ?? 1
is the length of perpendicular from the centre (0,0,0) of the sphere ?? 2
+?? 2
+?? 2
=?? 2
to the plane ?? +?? +?? =?? , then
?? 1
=|
0+0+0-?? v1+1+1
|=
?? v3
? radius of the circle (iii) is
?? 1
=v?? 2
-?? 1
2
=
v
?? 2
-
?? 2
3
=
v
2
3
??
Area of circle (iii) =?? ?? 1
2
=?? ·
2
3
?? 2
=
2?? 3
?? 2
Similarly area of circle (iv) is
2?? 3
?? 2
5.3 A variable plane is parallel to the plane
?? ?? +
?? ?? +
?? ?? =??
and meets the axes in ?? ,?? ,?? respectively. Prove that the circle ?????? lics on the
cone
???? (
?? ?? +
?? ?? )+???? (
?? ?? +
?? ?? )+???? (
?? ?? +
?? ?? )=??
(2012 : 20 Marks)
Solution:
The equation of any plane parallel to the given plane
?? ?? +
?? ?? +
?? ?? =0 is
?? ?? +
?? ?? +
?? ?? =?? (??)
It is given that the plane (i) meets the co-ordinate axes in ?? ,?? and ??
??? ,?? and ?? are (???? ,0,0),(0,???? ,0) and (0,0,???? ) respectively.
Equation of any sphere passing through the points ?? ,?? ,?? ,?? is
?? 2
+?? 2
+?? 2
-?????? -?????? -?????? =0
???? ?? 2
+?? 2
+?? 2
-?? (???? -???? -???? )=0 (???? )
The equation (i) and (ii) together represents the circle ?????? .
Eliminating ?? from (i) and (ii), the required cone is :
?? 2
+?? 2
+?? 2
-(
?? ?? +
?? ?? +
?? ?? )(???? +???? +???? )=0
or ???? (
?? ?? +
?? ?? )+???? (
?? ?? +
?? ?? )+???? (
?? ?? +
?? ?? )=0
Page 4
Edurev123
5. Cone and its properties
5.1 Prove that the normal from the point (?? ,?? ,?? ) to the paraboloid
?? ?? ?? ?? +
?? ?? ?? ?? =?? ?? be
on the cone
?? ?? -?? -
?? ?? -?? +
?? ?? -?? ?? ?? -?? =??
(Note: There is an error in the question of (+) sign instead of (-) before second
term.)
(2009 : 20 Marks)
Solution:
Approach : From the general equation of normal passing through a point (?? ,?? ,?? )
eliminate the direction cosines.
?? 2
?? 2
+
?? 2
?? 2
=2?? (??)
is given equation of paraboloid equation of tangent plane to paraboloid at (?? ,?? ,h) is
????
?? 2
+
????
?? 2
=(?? +h)
? Normal to paraboloid at (?? ,?? ,h) has direction cosines (
?? ?? 2
,
?? ?? 2
,-1) and the equation of
normal is
?? 2
(?? -?? )
?? =
?? 2
(?? -?? )
?? =
?? -h
-1
It passes through a point (?? ,?? ,?? ) if
?? 2
(?? -?? )
?? =
?? 2
(?? -?? )
?? =
?? -h
-1
=?? (let)
? ?? =
?? 2
?? ?? 2
+?? ;?? =
?? 2
?? ?? 2
+?? ;h=?? +??
Now let any normal through (?? ,?? ,?? ) be
?? -?? ?? =
?? -?? ?? =
?? -?? ?? (???? )
Then,
?? ?? /?? 2
=
?? ?? /?? 2
=
?? -1
(any normal must have such d.c.'s)
?
??(?? 2
+?? )
?? =
?? (?? 2
+?? )
?? =
?? -1
?
?? -1
=
?? 2
-?? 2
?? ?? -
?? ?? ? ?? ?? (
?? ?? -
?? ?? )=?? 2
-?? 2
? Replacing ??,?? ,?? from (ii)
?? ?? -?? -
?? ?? -?? +
?? 2
-?? 2
?? -?? =0
5.2 Show that the cone ???? +???? +???? =?? cuts the sphere ?? ?? +?? ?? +?? ?? =?? ?? in two
equal circles, and find their area.
(2011 : 20 Marks)
Solution:
The given equations are
?? 2
+?? 2
+?? 2
=?? 2
(??)
?????? ???? +???? +???? =0 (???? )
Multiply (ii) by 2 and add it to (i), we get
or
?? 2
+?? 2
+?? 2
+2(???? +2?? +???? ) =?? 2
(?? +?? +?? )
2
=?? 2
??? +?? +?? =±??
? The equations of the required circles are ·
?? 2
+?? 2
+?? 2
=?? 2
,?? +?? +?? =?? (?????? )
?? 2
+?? 2
+?? 2
=?? 2
,?? +?? +?? =-?? (???? )
Area of Circle (i)
Centre of the sphere ?? 2
+?? 2
+?? 2
=?? 2
is (0,0,0) .
If ?? 1
is the length of perpendicular from the centre (0,0,0) of the sphere ?? 2
+?? 2
+?? 2
=?? 2
to the plane ?? +?? +?? =?? , then
?? 1
=|
0+0+0-?? v1+1+1
|=
?? v3
? radius of the circle (iii) is
?? 1
=v?? 2
-?? 1
2
=
v
?? 2
-
?? 2
3
=
v
2
3
??
Area of circle (iii) =?? ?? 1
2
=?? ·
2
3
?? 2
=
2?? 3
?? 2
Similarly area of circle (iv) is
2?? 3
?? 2
5.3 A variable plane is parallel to the plane
?? ?? +
?? ?? +
?? ?? =??
and meets the axes in ?? ,?? ,?? respectively. Prove that the circle ?????? lics on the
cone
???? (
?? ?? +
?? ?? )+???? (
?? ?? +
?? ?? )+???? (
?? ?? +
?? ?? )=??
(2012 : 20 Marks)
Solution:
The equation of any plane parallel to the given plane
?? ?? +
?? ?? +
?? ?? =0 is
?? ?? +
?? ?? +
?? ?? =?? (??)
It is given that the plane (i) meets the co-ordinate axes in ?? ,?? and ??
??? ,?? and ?? are (???? ,0,0),(0,???? ,0) and (0,0,???? ) respectively.
Equation of any sphere passing through the points ?? ,?? ,?? ,?? is
?? 2
+?? 2
+?? 2
-?????? -?????? -?????? =0
???? ?? 2
+?? 2
+?? 2
-?? (???? -???? -???? )=0 (???? )
The equation (i) and (ii) together represents the circle ?????? .
Eliminating ?? from (i) and (ii), the required cone is :
?? 2
+?? 2
+?? 2
-(
?? ?? +
?? ?? +
?? ?? )(???? +???? +???? )=0
or ???? (
?? ?? +
?? ?? )+???? (
?? ?? +
?? ?? )+???? (
?? ?? +
?? ?? )=0
5.4 A cone has for its guiding curve the circle ?? ?? +?? ?? +?? ???? +?? ???? =?? ,?? =?? and
passes through a fixed point (?? ,?? ,?? ) . If the section of the cone by the plane ?? =??
is a rectangular hyperbola, prove that the vertex lies on the fixed circle.
?? ?? +?? ?? +?? ?? +?? ???? +?? ???? =?? ?? ???? +?? ???? +???? =??
(2013 : 15 Marks)
Solution:
Let ?? (?? ,?? ,?? ) be the vertex.
Any line through ??
?? -?? ?? =
?? -?? ?? =
?? -?? ?? (??)
It passes through ?? =0
?
?? -?? ?? =
?? -?? ?? =
-?? ??
? ?? =
-????
?? +?? ·?? =(
-????
?? +?? )
? Point of intersection with ?? =0
(?? -
????
?? ,?? -
????
?? ,0)
It lies on ?? 2
+?? 2
+2???? +2???? =0
? (?? -
????
?? )
2
+(?? -
????
?? )
2
+2?? (?? -
????
?? )+2?? (?? -
????
?? )=0
? (???? -???? )
2
+(???? -???? )2+2???? (???? -???? )+2???? (???? -???? )=0
Eliminating ??,?? ,?? from (i)
[?? (?? -?? )-?? (?? -?? )]
2
+[?? (?? -?? )-?? (?? -?? )]
2
+2?? (?? -?? )[?? (?? -?? )-(?? -?? )?? ]+2?? (?? -?? )[?? (?? -?? )-(?? -?? )?? ]=0
? (???? -???? )
2
+(???? -???? )
2
+2?? (?? -?? )(???? -???? )+2?? (?? -?? )(???? -???? )=0 (???? )
Intersection with ?? =0 of (ii)
(???? -???? )
2
+(???? )
2
+2?? (?? -?? )(???? -???? )+2?? (?? -?? )(???? )=0
This is rectangular hyperbola if
Page 5
Edurev123
5. Cone and its properties
5.1 Prove that the normal from the point (?? ,?? ,?? ) to the paraboloid
?? ?? ?? ?? +
?? ?? ?? ?? =?? ?? be
on the cone
?? ?? -?? -
?? ?? -?? +
?? ?? -?? ?? ?? -?? =??
(Note: There is an error in the question of (+) sign instead of (-) before second
term.)
(2009 : 20 Marks)
Solution:
Approach : From the general equation of normal passing through a point (?? ,?? ,?? )
eliminate the direction cosines.
?? 2
?? 2
+
?? 2
?? 2
=2?? (??)
is given equation of paraboloid equation of tangent plane to paraboloid at (?? ,?? ,h) is
????
?? 2
+
????
?? 2
=(?? +h)
? Normal to paraboloid at (?? ,?? ,h) has direction cosines (
?? ?? 2
,
?? ?? 2
,-1) and the equation of
normal is
?? 2
(?? -?? )
?? =
?? 2
(?? -?? )
?? =
?? -h
-1
It passes through a point (?? ,?? ,?? ) if
?? 2
(?? -?? )
?? =
?? 2
(?? -?? )
?? =
?? -h
-1
=?? (let)
? ?? =
?? 2
?? ?? 2
+?? ;?? =
?? 2
?? ?? 2
+?? ;h=?? +??
Now let any normal through (?? ,?? ,?? ) be
?? -?? ?? =
?? -?? ?? =
?? -?? ?? (???? )
Then,
?? ?? /?? 2
=
?? ?? /?? 2
=
?? -1
(any normal must have such d.c.'s)
?
??(?? 2
+?? )
?? =
?? (?? 2
+?? )
?? =
?? -1
?
?? -1
=
?? 2
-?? 2
?? ?? -
?? ?? ? ?? ?? (
?? ?? -
?? ?? )=?? 2
-?? 2
? Replacing ??,?? ,?? from (ii)
?? ?? -?? -
?? ?? -?? +
?? 2
-?? 2
?? -?? =0
5.2 Show that the cone ???? +???? +???? =?? cuts the sphere ?? ?? +?? ?? +?? ?? =?? ?? in two
equal circles, and find their area.
(2011 : 20 Marks)
Solution:
The given equations are
?? 2
+?? 2
+?? 2
=?? 2
(??)
?????? ???? +???? +???? =0 (???? )
Multiply (ii) by 2 and add it to (i), we get
or
?? 2
+?? 2
+?? 2
+2(???? +2?? +???? ) =?? 2
(?? +?? +?? )
2
=?? 2
??? +?? +?? =±??
? The equations of the required circles are ·
?? 2
+?? 2
+?? 2
=?? 2
,?? +?? +?? =?? (?????? )
?? 2
+?? 2
+?? 2
=?? 2
,?? +?? +?? =-?? (???? )
Area of Circle (i)
Centre of the sphere ?? 2
+?? 2
+?? 2
=?? 2
is (0,0,0) .
If ?? 1
is the length of perpendicular from the centre (0,0,0) of the sphere ?? 2
+?? 2
+?? 2
=?? 2
to the plane ?? +?? +?? =?? , then
?? 1
=|
0+0+0-?? v1+1+1
|=
?? v3
? radius of the circle (iii) is
?? 1
=v?? 2
-?? 1
2
=
v
?? 2
-
?? 2
3
=
v
2
3
??
Area of circle (iii) =?? ?? 1
2
=?? ·
2
3
?? 2
=
2?? 3
?? 2
Similarly area of circle (iv) is
2?? 3
?? 2
5.3 A variable plane is parallel to the plane
?? ?? +
?? ?? +
?? ?? =??
and meets the axes in ?? ,?? ,?? respectively. Prove that the circle ?????? lics on the
cone
???? (
?? ?? +
?? ?? )+???? (
?? ?? +
?? ?? )+???? (
?? ?? +
?? ?? )=??
(2012 : 20 Marks)
Solution:
The equation of any plane parallel to the given plane
?? ?? +
?? ?? +
?? ?? =0 is
?? ?? +
?? ?? +
?? ?? =?? (??)
It is given that the plane (i) meets the co-ordinate axes in ?? ,?? and ??
??? ,?? and ?? are (???? ,0,0),(0,???? ,0) and (0,0,???? ) respectively.
Equation of any sphere passing through the points ?? ,?? ,?? ,?? is
?? 2
+?? 2
+?? 2
-?????? -?????? -?????? =0
???? ?? 2
+?? 2
+?? 2
-?? (???? -???? -???? )=0 (???? )
The equation (i) and (ii) together represents the circle ?????? .
Eliminating ?? from (i) and (ii), the required cone is :
?? 2
+?? 2
+?? 2
-(
?? ?? +
?? ?? +
?? ?? )(???? +???? +???? )=0
or ???? (
?? ?? +
?? ?? )+???? (
?? ?? +
?? ?? )+???? (
?? ?? +
?? ?? )=0
5.4 A cone has for its guiding curve the circle ?? ?? +?? ?? +?? ???? +?? ???? =?? ,?? =?? and
passes through a fixed point (?? ,?? ,?? ) . If the section of the cone by the plane ?? =??
is a rectangular hyperbola, prove that the vertex lies on the fixed circle.
?? ?? +?? ?? +?? ?? +?? ???? +?? ???? =?? ?? ???? +?? ???? +???? =??
(2013 : 15 Marks)
Solution:
Let ?? (?? ,?? ,?? ) be the vertex.
Any line through ??
?? -?? ?? =
?? -?? ?? =
?? -?? ?? (??)
It passes through ?? =0
?
?? -?? ?? =
?? -?? ?? =
-?? ??
? ?? =
-????
?? +?? ·?? =(
-????
?? +?? )
? Point of intersection with ?? =0
(?? -
????
?? ,?? -
????
?? ,0)
It lies on ?? 2
+?? 2
+2???? +2???? =0
? (?? -
????
?? )
2
+(?? -
????
?? )
2
+2?? (?? -
????
?? )+2?? (?? -
????
?? )=0
? (???? -???? )
2
+(???? -???? )2+2???? (???? -???? )+2???? (???? -???? )=0
Eliminating ??,?? ,?? from (i)
[?? (?? -?? )-?? (?? -?? )]
2
+[?? (?? -?? )-?? (?? -?? )]
2
+2?? (?? -?? )[?? (?? -?? )-(?? -?? )?? ]+2?? (?? -?? )[?? (?? -?? )-(?? -?? )?? ]=0
? (???? -???? )
2
+(???? -???? )
2
+2?? (?? -?? )(???? -???? )+2?? (?? -?? )(???? -???? )=0 (???? )
Intersection with ?? =0 of (ii)
(???? -???? )
2
+(???? )
2
+2?? (?? -?? )(???? -???? )+2?? (?? -?? )(???? )=0
This is rectangular hyperbola if
Coefficient of ?? 2
+ Coefficieni of ?? 2
=0
? ?? 2
+?? 2
+?? 2
+2???? +2???? =0 (?????? )
(ii) passes through fixed point (0,0,?? )
? (???? )
2
+(???? )
2
+2?? (?? -?? )???? +2?? (?? -?? )???? =0
? (?? 2
+?? 2
+2???? +2???? )?? 2
-2???????? -2???????? =0 (???? )
Using (ii), (iii) & (iv) are equivalent to
-?? 2
?? 2
-2(???? +???? )???? =0
???? (2???? +2???? +???? ) =0
? (2???? +2???? +???? )=0 (?? )
as ???? is not identically zero.
? (iii) and (iv) are required conditions.
Locus of ?? (?? ,?? ,?? ) is
?? 2
+?? 2
+?? 2
+2???? +2???? =0
2???? +2???? +???? =0
5.5 Examine whether the plane ?? +?? +?? =?? cuts the cone ???? +???? +???? =?? in
perpendicular lines.
(2014 : 10 Marks)
Solution:
From the equation of plane and the cone it is clear that the lines of intensities passes
through origin.
Let equation of lines be
?? ?? =
?? ?? =
?? ?? (??)
(i) must satisfy equation of plane and cone.
So,
?? +?? +?? =0 (???? )
and ???? +???? +???? =0
From (ii) and (iii)
???? +(?? +?? )×-(?? +?? )=0
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Nov 21, 2024 Last updated |
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Additional Information about Cone and its properties for UPSC Preparation
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The content of Cone and its properties is prepared as per the latest UPSC syllabus.
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