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Byju's Answer
Standard XII
Mathematics
Normal of a Curve y = f(x)
The different...
Question
The differential equation satisfied by the curves
e
2
y
+
2
b
x
e
y
+
b
2
=
0
,
where
b
is a parameter, is
A
(
1
+
x
2
)
d
2
y
d
x
2
=
0
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B
(
x
2
−
1
)
(
d
y
d
x
)
2
−
1
=
0
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C
(
x
2
−
1
)
d
2
y
d
x
2
+
2
=
0
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D
(
x
2
−
x
)
d
y
d
x
−
y
2
=
0
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Solution
The correct option is
B
(
x
2
−
1
)
(
d
y
d
x
)
2
−
1
=
0
e
2
y
+
2
b
x
e
y
+
b
2
=
0
⇒
e
2
y
+
2
b
x
e
y
+
b
2
x
2
=
b
2
x
2
−
b
2
⇒
(
e
y
+
b
x
)
2
=
b
2
(
x
2
−
1
)
⇒
e
y
+
b
x
=
±
b
√
x
2
−
1
⇒
e
y
=
−
b
(
x
±
√
x
2
−
1
)
⋯
(
1
)
Differentiating equation
(
1
)
with respect to
x
,
we get
e
y
d
y
d
x
=
−
b
(
√
x
2
−
1
±
x
)
√
x
2
−
1
⋯
(
2
)
From equations
(
1
)
and
(
2
)
,
√
x
2
−
1
d
y
d
x
=
±
1
⇒
(
x
2
−
1
)
(
d
y
d
x
)
2
=
1
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