The angleof rotationof axes to remove xy term in the equation
Answers
Ax/ndjs(9(9794949
Step-by-step explanation:
The equation given here
x
2
−
4
x
y
+
4
y
2
+
5
√
5
+
1
=
0
is of the form
A
x
2
+
B
x
y
+
C
y
2
+
D
x
+
E
y
+
F
=
0
.
What a rotation does is it changes
x
&
y
-axes to
x
'
&
y
'
-axes, as shown below,.
enter image source here
In such a case, the relation between coordinate
(
x
,
y
)
and new coordinates
(
x
'
,
y
'
)
is given by
x
=
x
'
cos
θ
−
y
'
sin
θ
and
y
=
x
'
sin
θ
+
y
'
cos
θ
and reverse is
x
'
=
x
cos
θ
+
y
sin
θ
and
y
'
=
−
x
sin
θ
−
cos
θ
Note that latter equations are equivalent to rotation by
−
θ
.
In such cases, we can eliminate
x
y
if rotated by
θ
=
A
−
C
B
In given equation we have
cot
2
θ
=
1
−
4
−
4
=
3
4
i.e.
cot
2
θ
−
1
2
cot
θ
=
3
4
or
4
cot
2
θ
−
6
cot
θ
−
4
=
0
or
(
2
cot
θ
−
4
)
(
2
cot
θ
+
1
)
=
0
i.e.
cot
θ
=
2
or
−
1
2
These two angles relate to
θ
and
θ
−
90
o
in the image above. Working out for
cot
θ
=
2
Hence, either
sin
θ
=
1
√
5
and
cos
θ
=
2
√
5
or
sin
θ
=
−
2
√
5
and
cos
θ
=
1
√
5
and we have
x
=
2
x
'
√
5
+
y
'
√
5
and
y
=
x
'
√
5
+
2
y
'
√
5
and putting these in given equation and simplifying we get
9
y
2
+
25
√
5
+
5
=
0
One can also try for
cot
θ
=
−
1
2
, for which we get
sin
θ
=
−
2
√
5
and
cos
θ
=
1
√
5
x
'
=
x
cos
θ
+
y
sin
θ
and
y
'
=
−
x
sin
θ
−
cos
θ
i.e
x
'
=
x
√
5
−
2
y
√
5
and
y
'
=
2
x
√
5
−
y
√
5
and simplifying
9
x
2
+
25
√
5
=
5
=
0
Note - Please observe that above equation
x
2
−
4
x
y
+
4
y
2
+
5
√
5
+
1
=
0
⇔
(
x
−
2
y
)
2
+
+
5
√
5
+
1
=
0
and as LHS for
x
∈
R
and
y
∈
R
is always positive, does not have real solution and as such cannot be represented on Cartesian Plane.