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2 1 2 1

2 1 2 1

1 1

1 1

or,

2 2

2 2

    

    

  

  

p p p p

p p p p

x x

x x

[by componendo - dividendo]

1 0.

2

2

2 or,^1

1 1

or,

2

2

2

?  





x

p

p

x

p

p p

p

x

Example 8. If ( )

3 3

aa b

a b c

a b



 



, prove that a,b,c are ordered proportional.

Solution : Given that, ( )

3 3

aa b

a b c

a b



 



or, ( )

3 3

aa b

a b c

a b



 



( )

( )( )

or,

2 2

aa b

a b c

a b a ab b



 

  

a

a b c

a ab b

 

(^2)   2
or, [Dividing both sides by (a +b)]
or, a^2 abb^2 a^2 abac
? b^2 ac
?a,b,c are ordered proportional.
Example 9. If
d a
c d
b c
a b




, prove that c a or abcd 0.
Solution : Given that,
d a
c d
b c
a b




or, 1  1





d a
c d
b c
a b
[subtracting 1 from both the sides]
d a
c d d a
b c
a b b c

  

  
or,
d a
c a
b c
a c




or,
or 0





d a
a c
b c
a c
,
0
1 1
or, ( ) ̧
¹
·
̈
©
§




b c d a
a c