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Byju's Answer
Standard XII
Mathematics
Continuity in an Interval
Prove that, i...
Question
Prove that, if
a
,
b
,
c
and
d
be positive rationals such that,
a
+
√
b
=
c
+
√
d
,
then either
a
=
c
and
b
=
d
or
b
and
d
are squares of rationals.
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Solution
If
a
=
c
, then
a
+
√
b
=
c
+
√
d
⇒
√
b
=
√
d
⇒
b
=
d
So, let
a
≠
c
. Then, there exists a positive rational number
x
such that
a
=
c
+
x
.
Now,
⇒
a
+
√
b
=
c
+
√
d
⇒
c
+
x
+
√
b
=
c
+
√
d
[
∵
a
=
c
+
x
]
⇒
x
+
√
b
=
√
d
⇒
(
x
+
√
b
)
2
=
(
√
d
)
2
⇒
x
2
+
2
√
b
x
+
b
=
d
⇒
√
b
=
d
−
x
2
−
b
2
x
⇒
√
b
is rational
[
∴
d
,
x
,
b
a
r
e
r
a
t
i
o
n
a
l
s
∴
d
−
x
2
−
b
2
2
x
i
s
r
a
t
i
o
n
a
l
]
⇒
b
is the square of a rational number.
From
(
i
)
, we have
√
d
=
x
+
√
b
⇒
√
d
is rational
⇒
d
is the square of a rational number.
Hence, either
a
=
c
and
b
=
d
or
b
and
d
are the squares of rationals.
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0
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