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NCERT Class XI Mathematics - Sequences and Series - Solutions

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Question : 97
Total: 106
The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show that a : b = (m+√m2−n2) : (m−√m2−n2)
Solution:  
We have
a+b
2
√ab
 =
m
n
 ⇒
a+b
2√ab
 =
m
n
Applying componendo and dividendo, we get
a+b+2√ab
a+b−2√ab
 =
m+n
m−n
 ⇒
(√a+√b)2
(√a−√b)2
 =
m+n
m−n
 ⇒
√a+√b
√a−√b
 =
√m+n
√m−n
Again applying componendo and dividendo, we get
√a+√b+√a−√b
√a+√b−√a+√b
 =
√m+n+√m−n
√m+n−√m−n
 ⇒
√a
√b
 =
√m+n+√m−n
√m+n−√m−n
Squaring on both sides, we get
a
b
 =
m+n+m−n+2√m2−n2
m+n+m−n−2√m2−n2
⇒
a
b
 =
2m+2√m2−n2
2m−2√m2−n2
 ⇒
a
b
 =
m+√m2−n2
m−√m2−n2
⇒ a : b = (m+√m2−n2) : (m−√m2−n2)
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