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Page 1
Edurev123
Analytic Geometry
1. Straight lines
1.1 A line is drawn through a variable point on the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? =?? ,?? =?? to meet
fixed lines ?? =???? , ?? =?? and ?? =-???? ,?? =-?? . Find the locus of the line.
(2009 : 12 Marks)
Solution:
Approach: Use general equation of line intersecting two lines given in planar form.
Given fixed lines are
?? -???? =0,?? -0=0 (??)
?? +???? =0,?? +?? =0 (???? )
General equation of line intersecting both
(?? -???? )+?? 1
(?? -?? )=0=(?? +???? )+?? 2
(?? +?? ) (?????? )
If it meets ellipse we eliminate ?? 1
and ?? 2
Putting ?? =0 in (iii)
?? -???? -?? 1
?? =0;?? +???? +?? 2
?? =0
?
?? -?? 2
?? +?? 1
?? =
?? -(?? 1
+?? 2
)
=
?? 2??
? ?? =
-(?? 1
+?? 2
)?? 2?? ;?? =
(?? 1
-?? 2
)?? 2
Putting this in equation of ellipse
(?? 1
+?? 2
)
2
?? 2
4?? 2
?? 2
+
(?? 1
-?? 2
)
2
?? 2
4?? 2
=1
(?? 1
+?? 2
)
2
?? 2
?? 2
+(?? 1
-?? 2
)
2
?? 2
?? 2
?? 2
=4?? 2
?? 2
?? 2
Substituting ?? 1
and ?? 2
from (iii)
{(
???? -?? ?? -?? )+(-
???? +?? ?? +?? )}
2
?? 2
?? 2
+{(
???? -?? ?? -?? )+(
???? +?? ?? +?? )}
2
×?? 2
?? 2
=4?? 2
?? 2
?? 2
Page 2
Edurev123
Analytic Geometry
1. Straight lines
1.1 A line is drawn through a variable point on the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? =?? ,?? =?? to meet
fixed lines ?? =???? , ?? =?? and ?? =-???? ,?? =-?? . Find the locus of the line.
(2009 : 12 Marks)
Solution:
Approach: Use general equation of line intersecting two lines given in planar form.
Given fixed lines are
?? -???? =0,?? -0=0 (??)
?? +???? =0,?? +?? =0 (???? )
General equation of line intersecting both
(?? -???? )+?? 1
(?? -?? )=0=(?? +???? )+?? 2
(?? +?? ) (?????? )
If it meets ellipse we eliminate ?? 1
and ?? 2
Putting ?? =0 in (iii)
?? -???? -?? 1
?? =0;?? +???? +?? 2
?? =0
?
?? -?? 2
?? +?? 1
?? =
?? -(?? 1
+?? 2
)
=
?? 2??
? ?? =
-(?? 1
+?? 2
)?? 2?? ;?? =
(?? 1
-?? 2
)?? 2
Putting this in equation of ellipse
(?? 1
+?? 2
)
2
?? 2
4?? 2
?? 2
+
(?? 1
-?? 2
)
2
?? 2
4?? 2
=1
(?? 1
+?? 2
)
2
?? 2
?? 2
+(?? 1
-?? 2
)
2
?? 2
?? 2
?? 2
=4?? 2
?? 2
?? 2
Substituting ?? 1
and ?? 2
from (iii)
{(
???? -?? ?? -?? )+(-
???? +?? ?? +?? )}
2
?? 2
?? 2
+{(
???? -?? ?? -?? )+(
???? +?? ?? +?? )}
2
×?? 2
?? 2
=4?? 2
?? 2
?? 2
?[(???? -?? )(?? +?? )-(???? +?? )(?? -?? )]
2
?? 2
?? 2
+[(???? -?? )(?? +?? )+(???? +?? )(?? -?? )]
2
?? 2
?? 2
?? 2
=4?? 2
?? 2
?? 2
(?? 2
-?? 2
)
2
?[?????? -???? ]
2
?? 2
?? 2
+[?????? -???? ]
2
?? 2
?? 2
?? 2
=?? 2
?? 2
?? 2
(?? 2
-?? 2
)
2
which is required locus.
1.2 Prove that two of the straight lines represented by the equation
?? ?? +?? ?? ?? ?? +???? ?? ?? +?? ?? =??
will be at right angles, if ?? +?? =-?? .
(2012 : 12 Marks)
Solution:
The given equation is a homogeneous equation of third degree and hence it represents
three straight lines through the origin.
Let ?? =???? be any of these lines.
Replacing
?? ?? by min?? 3
+?? ?? 2
?? +???? ?? 2
+?? 3
=0 or 1+?? ?? ?? +?? ?? 2
?? 2
+
?? 3
?? 3
=0, we get
?? 3
+?? ?? 2
+???? +1=0 (??)
Let ?? 1
,?? 2
,?? 3
be its roots, then
?? 1
·?? 2
·?? 3
=-1
But, two of these limes, say with slopes, ?? 1
and ?? 2
, are at right angles,
then, ?? 1
·?? 2
=-1
Thus, (-?? 3
)=1 ???? ?? 3
=1
But ?? 3
is a root of (i)
? 1+?? +?? +1=0
???? ?? +?? =-2
1.3 Verify if the lines
?? -?? +?? ?? -?? =
?? -?? ?? =
?? -?? -?? ?? +?? and
?? -?? +?? ?? -?? =
?? -?? ?? =
?? -?? -?? ?? +?? are coplanar. If yes,
then find the equation of the plane in which they lie?
(2014: 7 Marks)
Solution:
Two straight lines
Page 3
Edurev123
Analytic Geometry
1. Straight lines
1.1 A line is drawn through a variable point on the ellipse
?? ?? ?? ?? +
?? ?? ?? ?? =?? ,?? =?? to meet
fixed lines ?? =???? , ?? =?? and ?? =-???? ,?? =-?? . Find the locus of the line.
(2009 : 12 Marks)
Solution:
Approach: Use general equation of line intersecting two lines given in planar form.
Given fixed lines are
?? -???? =0,?? -0=0 (??)
?? +???? =0,?? +?? =0 (???? )
General equation of line intersecting both
(?? -???? )+?? 1
(?? -?? )=0=(?? +???? )+?? 2
(?? +?? ) (?????? )
If it meets ellipse we eliminate ?? 1
and ?? 2
Putting ?? =0 in (iii)
?? -???? -?? 1
?? =0;?? +???? +?? 2
?? =0
?
?? -?? 2
?? +?? 1
?? =
?? -(?? 1
+?? 2
)
=
?? 2??
? ?? =
-(?? 1
+?? 2
)?? 2?? ;?? =
(?? 1
-?? 2
)?? 2
Putting this in equation of ellipse
(?? 1
+?? 2
)
2
?? 2
4?? 2
?? 2
+
(?? 1
-?? 2
)
2
?? 2
4?? 2
=1
(?? 1
+?? 2
)
2
?? 2
?? 2
+(?? 1
-?? 2
)
2
?? 2
?? 2
?? 2
=4?? 2
?? 2
?? 2
Substituting ?? 1
and ?? 2
from (iii)
{(
???? -?? ?? -?? )+(-
???? +?? ?? +?? )}
2
?? 2
?? 2
+{(
???? -?? ?? -?? )+(
???? +?? ?? +?? )}
2
×?? 2
?? 2
=4?? 2
?? 2
?? 2
?[(???? -?? )(?? +?? )-(???? +?? )(?? -?? )]
2
?? 2
?? 2
+[(???? -?? )(?? +?? )+(???? +?? )(?? -?? )]
2
?? 2
?? 2
?? 2
=4?? 2
?? 2
?? 2
(?? 2
-?? 2
)
2
?[?????? -???? ]
2
?? 2
?? 2
+[?????? -???? ]
2
?? 2
?? 2
?? 2
=?? 2
?? 2
?? 2
(?? 2
-?? 2
)
2
which is required locus.
1.2 Prove that two of the straight lines represented by the equation
?? ?? +?? ?? ?? ?? +???? ?? ?? +?? ?? =??
will be at right angles, if ?? +?? =-?? .
(2012 : 12 Marks)
Solution:
The given equation is a homogeneous equation of third degree and hence it represents
three straight lines through the origin.
Let ?? =???? be any of these lines.
Replacing
?? ?? by min?? 3
+?? ?? 2
?? +???? ?? 2
+?? 3
=0 or 1+?? ?? ?? +?? ?? 2
?? 2
+
?? 3
?? 3
=0, we get
?? 3
+?? ?? 2
+???? +1=0 (??)
Let ?? 1
,?? 2
,?? 3
be its roots, then
?? 1
·?? 2
·?? 3
=-1
But, two of these limes, say with slopes, ?? 1
and ?? 2
, are at right angles,
then, ?? 1
·?? 2
=-1
Thus, (-?? 3
)=1 ???? ?? 3
=1
But ?? 3
is a root of (i)
? 1+?? +?? +1=0
???? ?? +?? =-2
1.3 Verify if the lines
?? -?? +?? ?? -?? =
?? -?? ?? =
?? -?? -?? ?? +?? and
?? -?? +?? ?? -?? =
?? -?? ?? =
?? -?? -?? ?? +?? are coplanar. If yes,
then find the equation of the plane in which they lie?
(2014: 7 Marks)
Solution:
Two straight lines
?? -?? 1
?? 1
=
?? -?? 1
?? 1
=
?? -?? 1
?? 1
and
?? -?? 2
?? 2
=
?? -?? 2
?? 2
=
?? -?? 2
?? 2
are coplanar if
|
?? 2
-?? 1
?? 2
-?? 1
?? 2
-?? 1
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
|=0
And equation of plane containing them, is
|
?? -?? 1
?? -?? 1
?? -?? 1
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
|=0
Here, in our case,
|
(?? -?? )-(?? -?? ) ?? -?? ?? +?? -(?? +?? )
?? -?? ?? ?? +?? ?? -?? ?? ?? +?? |
?? 1
??? 1
-?? 2
?? 3
??? 3
-?? 2
=|
?? -?? ?? -?? ?? -?? -?? ?? ?? -?? ?? ?? |=0 as ?? 1
=-?? 3
Hence, the given lines are coplanar.
The equation of the plane containing thein, is
|
?? -(?? -?? ) ?? -?? ?? -(?? +?? )
?? -?? ?? ?? +?? ?? -?? ?? ?? +?? |=0. Applying
?? 1
??? 1
-?? 2
?? 3
??? 3
-?? 2
|
?? -?? +?? ?? -?? ?? -?? -?? -?? ?? ?? -?? ?? ?? |=0?|
?? -2?? +?? ?? -?? ?? -?? -?? 0 ?? ?? 0 ?? ?? |=0 as ?? 1
??? 1
+?? 3
? ?? -2?? +?? =0
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FAQs on Straight lines - Mathematics Optional Notes for UPSC
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