Equation of a pair of straight lines passing through the origin
y
=
m
1
x
,
y
=
m
2
x
(
y
-
m
1
x
)(
y
-
m
2
x
) = 0
i.e.
m
1
m
2
x
2
-
(
m
1
+
m
2
)
xy
+
y
2
= 0
---
(**)
-- a homogenous equation of the second degree in
x
and
y
The standard form of a pair of straight lines is
ax
2
+ 2
hxy
+
by
2
= 0
---
(*)
If
(*)
and
(**)
are equivalent
if
m
1
m
2
=
and
m
1
+
m
2
=
If the lines are perpendicular, then
m
1
m
2
= 1
i.e.
a
+
b
= 0
And the angle
between the two lines is given by
tan
=
=
=
=
The two lines are the same
h
2
-
ab
= 0
perpendicular
a
+
b
= 0
Consider
ax
2
+ 2
hxy
+
by
2
= 0
---
(*)
when
b
0
=
y
=
i.e.
The equation of the two straight lines passing through the origin.
(I)
If
b =
0
and
a =
0
and
h
0
,
(*)
2
hy
= 0
which gives the lines
x
= 0
and
y
= 0
(II)
If
b =
0
and
a
0
, then
(*)
becomes
ax
2
+ 2
hxy
= 0
x
(
ax
+ 2
hy
) = 0
and represents the lines
x =
0
and
ax
+ 2
hy=
0
(III)
If
b
0
, then the two lines are real if
(*)
2
hy
= 0
which gives the lines
x
= 0
and
y
= 0
If
b
0
, then the two lines are
real if
h
2
-
ab
> 0
i.e.
h
2
>
ab
coincident if
h
2
=
ab
the point
(0,0)
OR
imaginary if
h
2
-
ab
< 0
i.e.
h
2
<
ab
Related Examples
Jump to page......
Revision
Angular bisectors
Condition for the general equation of the second degree in x and y to represent two straight lines
and m1
+ m2 = 
between
the two lines is given by
The two lines are the same
0
