If
then find dy/dx
Answers
Answered by
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Observe that,
√
1
−
x
2
and
√
1
−
y
2
are Meaningful, iff,
|
x
|
≤
1
,
and
,
|
y
|
≤
1
...
...
...
...
...
(
⋆
1
)
.
This means that, there is no Harm if we let,
x
=
sin
θ
,
and
,
y
=
sin
ϕ
...
...
...
...
...
...
.
.
(
⋆
2
)
.
∴
√
1
−
x
2
+
√
1
−
y
2
=
a
(
x
−
y
)
,
becomes,
cos
θ
+
cos
ϕ
=
a
(
sin
θ
−
sin
ϕ
)
.
∴
2
cos
(
θ
+
ϕ
2
)
cos
(
θ
−
ϕ
2
)
=
2
a
{
cos
(
θ
+
ϕ
2
)
sin
(
θ
−
ϕ
2
)
}
.
∴
cos
(
θ
−
ϕ
2
)
=
a
sin
(
θ
−
ϕ
2
)
.
cot
(
θ
−
ϕ
2
)
=
a
.
(
θ
−
ϕ
2
)
=
a
r
c
cot
a
.
∴
θ
−
ϕ
=
2
a
r
c
cot
a
.
From,
(
⋆
1
)
,
and
,
(
⋆
2
)
,
we have,
a
r
c
sin
x
−
a
r
c
sin
y
=
2
a
r
c
cot
a
.
∴
d
d
x
{
a
r
c
sin
x
−
a
r
c
sin
y
}
=
d
d
x
{
2
a
r
c
cot
a
}
.
∴
1
√
1
−
x
2
−
1
√
1
−
y
2
⋅
d
d
x
{
y
}
=
0
,
...
[The Chain Rule].
∴
1
√
1
−
x
2
−
1
√
1
−
y
2
⋅
d
y
d
x
=
0
.
∴
d
y
d
x
=
√
1
−
y
2
√
1
−
x
2
.
⇒
d
y
d
x
=
√
1
−
y
2
1
−
x
2
I hope this will helps you
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