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if a(y+z)=x , b(z+x)=y , c(x+y)=z prove that x^2/a(1-bc)=y^2/b(1-ca)=z^2/c(1-ab)

if a(y+z)=x , b(z+x)=y , c(x+y)=z prove that x^2/a(1-bc)=y^2/b(1-ca)=z^2/c(1-ab)

Grade:10

1 Answers

Arun
25750 Points
5 years ago
A(y+z)=b(z+x)=c(x+y) 
Assume n= a(y+z)=b(z+x)=c(x+y) 
n=a(y+z) ,n= b(z+x) , n= c(x+y) 
y+z = (n/a).....................(1) 
z+x = (n/b).....................(2) 
x+y = (n/c).....................(3) 
subtracting (2) - (1) , (3) - (2) , (1) - (3) we get 
x - y = n( a - b )/ab ..................(4) 
y - z = n ( b - c )/bc...................(5) 
z - x = n( c - a )/ac...................(6) 

x-y/c(a-b) = [ n ( a - b )/ab ] / c( a - b ) 
= n/abc 
y-z/a(b-c) = [ n ( b - c )/bc ] / a( b - c ) 
= n/abc 
z-x/b(c-a) = [ n ( c - a )/ac ] / b( c - a ) 
= n/abc 
y-z/a(b-c)=z-x/b(c-a)=x-y/c(a-b)
 
also
x^2/a(1-bc)=y^2/b(1-ca)=z^2/c(1-ab)

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