Question

If a(y+z) = b(z+x) = c(x+y) and out of a, b, c no two of them are equal then show that, y-z/a(b-c) = z-x/b(c-a) = x-y/c(a-b).

Answer 1

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)