PROBLEMS
02

LINE / LINES
Page
1
( 1 )
Find
the
parametric
equations
of
the
line
passing
through
A ( 3,
 2 )
and
B (  4,
5 )
and
hence
express
↔
AB,
→
AB
and
AB
as
sets.
≤
≤
+
≥
+
+
=
+
∈
∈
=
∈
∈
=
→
=
↔
=
↔
}
AB
and
}
AB
},
Further
.
R
t
,
1
t
0
)
2
7t
3,
7t
(
{
R
t
0,
t
)
2
7t
3,
7t
(
{
R
t
)
2
7t
3,
7t
(
{
AB
R
t
2,
7t
y
3,
7t
x
are
AB
of
equations
Parametric
:
Ans
l
l
l








( 2 )
If
the
length
of
the
perpendicular segment
from
the
origin
is
10
and
α
=

6
5
π
,
then
find
the
equation
of
the
line.
[ Ans:
3
x
+
y
+
20
=
0 ]
( 3 )
If
the
lines
3x
+
y
+
4
=
0,
3x
+
4y

15
=
0
and
24x

7y

3
=
0
contain
the
sides
of
a
triangle,
prove
that
the
triangle
is
isosceles.
( 4 )
Find
the
coordinates
of
the
point
at
a
distance
of
10
units
from
the
point
( 4,
 3 )
on
the
line
perpendicular
to
3x
+
4y
=
0.
[ Ans:
( 10,
5 ),
(  2,
 11 ) ]
( 5 )
A ( x
1
,
y
1
)
and
B ( x
2
,
y
2
)
are
points
of
the
plane.
If
the
line
ax
+
by
+
c
=
0
divides
AB,
find
the
ratio
in
which
it
divides
AB
from
A.
≠
+
+
+
+
+
+
:
λ
=

0
c
by
ax
,
c
by
ax
c
by
ax
1
:
Ans
2
2
2
2
1
1
( 6 )
If
the
sum
of
the
intercepts
on
the
axes
of
a
line
is
constant,
find
the
equation
satisfied
by
the
midpoint
of
the
segment
of
the
line
intercepted
between
the
axes.
[ Ans:
x
+
y
=
k,
where
2k
=
constant
sum
of
the
intercepts ]
( 7 )
Find
k
if
the
lines
kx

y

2
=
0,
2x
+
ky

5
=
0
and
4x

y

3
=
0
are
concurrent.
[ Ans:
k
=
3
or
 2 ]
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02

LINE / LINES
Page
2
( 8 )
Among
all
the
lines
passing
through
the
point
of
intersection
of
the
lines
x
+
y

7
=
0
and
4x

3y
=
0,
find
the
one
for
which
the
length
of
the
perpendicular
segment
on
it
from
the
origin
is
maximum.
[ Ans:
3x
+
4y

25
=
0 ]
( 9 )
Prove
that
the
product
of
the
perpendicular
distances
of
the
line
θ
sin
b
y
θ
cos
a
x
+
=
1
from
the
points
±
0
,
b
2
a
2

is
b
2
.
( 10 )
Prove
that
if
l
m
1
≠
l
1
m,
n
≠
n
1
and
l
2
+
m
2
=
l
1
2
+
m
1
2
,
then
the
lines
l
x
+
my
+
n
=
0,
l
1
x
+
m
1
y
+
n
1
=
0,
l
x
+
my
+
n
1
=
0
and
l
1
x
+
m
1
y
+
n
=
0
form
a
rhombus.
( 11 )
Prove
that
the
lines
( a
2

3b
2
) x
2
+
8abxy
+
( b
2

3a
2
) y
2
=
0
and
ax
+
by
+
c
=
0,
c
≠
0
contain
the
sides
of
an
equilateral
triangle
whose
area
is
)
b
a
(
3
c
2
2
2
+
.
( 12 )
Two
lines
are
represented
by
3x
2

7xy
+
2y
2

14x
+
13y
+
15
=
0.
Find
the
measure
of
the
angle
between
them
and
the
point
of
their
intersection.
π
5
4
,
5
7
,
4
:
Ans

( 13 )
If
the
intercepts
on
the
axes
by
the
line
x cos
α
+
y sin
α
=
p
are
a
and
b,
prove
that
a

2
+
b

2
=
p

2
.
( 14 )
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 Spring '06
 Jones
 Equations, Sets, Parametric Equations, triangle, Line segment

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