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Byju's Answer
Standard VI
Mathematics
Continued or Mean Proportion
If b is the...
Question
If
b
is the mean proportional between
a
and
c
, prove that
(
a
b
+
b
c
)
is the mean proportional between
(
a
2
+
b
2
)
and
(
b
2
+
c
2
)
.
Open in App
Solution
It is given that
b
is the mean proportional that is geometric mean between
a
and
c
.
⇒
b
2
=
a
c
⋯
⋯
(
1
)
And we have to prove
(
a
b
+
b
c
)
is the mean proportional between
(
a
2
+
b
2
)
and
(
b
2
+
c
2
)
that is we have to prove,
(
a
b
+
b
c
)
2
=
(
a
2
+
b
2
)
(
b
2
+
c
2
)
Consider
L
H
S
=
(
a
b
+
b
c
)
2
Expanding using formula
L
H
S
=
a
2
b
2
+
b
2
c
2
+
2
a
b
2
c
Using equation
(
1
)
L
H
S
=
a
2
(
a
c
)
+
a
c
(
c
)
2
+
2
a
.
a
c
.
c
=
a
3
c
+
a
c
3
+
2
a
2
c
2
Taking
a
c
as common
L
H
S
=
a
c
(
a
2
+
c
2
+
2
a
c
)
L
H
S
=
a
c
(
a
+
c
)
2
⋯
⋯
(
2
)
Now,
R
H
S
=
(
a
2
+
b
2
)
(
b
2
+
c
2
)
Using equation
(
1
)
R
H
S
=
(
a
2
+
a
c
)
(
a
c
+
c
2
)
Taking common terms out
R
H
S
=
a
(
a
+
c
)
c
(
a
+
c
)
R
H
S
=
a
c
(
a
+
c
)
2
⋯
⋯
(
3
)
From
(
2
)
and
(
3
)
L
H
S
=
R
H
S
Therefore it is proved that
(
a
b
+
b
c
)
is the mean proportional between
(
a
2
+
b
2
)
and
(
b
2
+
c
2
)
.
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Similar questions
Q.
If
b
is a mean proportional between
a
and
c
, prove that
a
2
−
b
2
+
c
2
a
−
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−
b
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+
c
−
2
=
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4
.
Q.
If
a
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,
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(i)
a
2
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a
b
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b
2
b
2
+
b
c
+
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2
=
a
c
(ii)
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b
2
+
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2
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+
b
+
c
)
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=
a
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b
+
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a
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+
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Q.
If a,b,c are in continued proportion prove that :
i)
a
2
+
a
b
+
b
2
b
2
+
b
c
+
c
2
=
a
c
.
ii)
a
4
+
a
2
b
2
+
b
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+
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2
c
2
+
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=
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2
.
Q.
If
a
,
b
a
n
d
c
in continued proportion, prove that:
a
2
+
a
b
+
b
2
b
2
+
b
c
+
c
2
=
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Q.
If
a
,
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,
c
,
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are in continued proportion, prove that
(
a
2
+
b
2
+
c
2
)
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