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Byju's Answer
Standard XII
Mathematics
Eccentric Angle : Ellipse
If θ is the...
Question
If
θ
is the acute angle between the lines represented by equation
a
x
2
+
2
h
x
y
+
b
y
2
=
0
, then prove that
tan
θ
=
∣
∣
∣
2
√
h
2
−
a
b
a
+
b
∣
∣
∣
,
a
+
b
≠
0
.
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Solution
Let
m
1
a
n
d
m
2
be the slopes of the lines by the equation
Given,
a
x
2
+
2
h
x
y
+
b
y
2
=
0
....... (i)
Let
y
=
m
1
x
and
y
=
m
2
x
∴
(
y
−
m
1
x
)
(
y
−
m
2
x
)
=
0
and
m
1
m
2
x
2
−
(
m
1
+
m
2
)
x
y
+
y
2
=
0
......... (ii)
Comparing (i) and (ii), we get
m
1
m
2
a
=
1
b
=
m
1
+
m
2
2
h
∴
m
1
m
2
=
a
b
and
m
1
+
m
2
=
−
2
h
b
Thus,
(
m
1
−
m
2
)
2
=
(
m
1
+
m
2
)
2
−
4
m
1
m
2
(
m
1
−
m
2
)
2
=
(
−
2
h
b
)
2
−
4
(
a
b
)
(
m
1
−
m
2
)
2
=
4
(
h
2
−
a
b
)
b
2
Let the angle between
y
=
m
1
x
and
y
=
m
2
x
be
θ
.
tan
θ
=
∣
∣
∣
m
1
−
m
2
1
+
m
1
m
2
∣
∣
∣
=
∣
∣ ∣
∣
√
m
1
−
m
2
2
1
+
m
1
m
2
∣
∣ ∣
∣
=
∣
∣ ∣ ∣ ∣ ∣ ∣
∣
√
4
(
h
2
−
a
b
)
b
2
1
+
a
b
∣
∣ ∣ ∣ ∣ ∣ ∣
∣
=
∣
∣ ∣
∣
2
√
(
h
2
−
a
b
)
a
+
b
∣
∣ ∣
∣
, if
a
+
b
≠
0
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0
Similar questions
Q.
If
θ
is the measure of the acute angle between the lines represented by the equation
a
x
2
+
2
h
x
y
+
b
y
2
=
0
, then prove that
tan
θ
=
∣
∣
∣
2
√
h
2
−
a
b
a
+
b
∣
∣
∣
where
a
+
b
≠
0
and
b
≠
0
.
Find the condition for coincident lines
Q.
The angle between the lines represented by the equation
a
x
2
+
2
h
x
y
+
b
y
2
=
0
is given by