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or, ( ) 0

 

  



b c d a

d a b c

a c

or, (ac)(dabc) 0

? Either ac 0 i.e.,a c
or, abcd 0.

Example 10. If
x y

z

z x

y

y z

x







and x,y,z are not mutually equal, prove

that the value of each ratio is either equal  1 or equal
2

1

.

Solution : Let, k
x y

z

z x

y

y z

x



?x k(yz)................(i)
y k(zx)...................(ii
z k(xy)...................(iii)
Subtracting (ii) from (i), we get,
xy k(yx) or, k(yx)  yx

?k  1
Again, adding (i),(ii) and (iii), we get,

2

1

2 ( )

1 ( )

k

( ) 2 ( )

 

 

?

        

x y z

x y z

x y z ky z z x x y kx y z

? Therefore, the value of each of the ratio is  1 or
2

1

.

Example 11. If ax by cz, show that.
2 2 2

2 2 2

c

ab

b

ca

a

bc

xy

z

zx

y

yz

x

   

Solution : Let, ax by cz k

c

k

z

b

k

y

a

k

?x , ,

Now,
2 2

2

2 2

2

2 2

2 2 2 2

k

ab

c

k

k

ca

b

k

k

bc

a

k

xy

z

zx

y

yz

x

  u  u  u

(^22) c 2
ab
b
ca
a
bc
 
i.e.,.
2 2 2
2 2 2
c
ab
b
ca
a
bc
xy
z
zx
y
yz
x