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if y+z/pb+qc=z+x/pc+qa=x+y/pa+qb,show that 2(x+y+z)/a+b+c=(b+c)x+(c+a)y+(a+b)z/bc+ca+ab

if y+z/pb+qc=z+x/pc+qa=x+y/pa+qb,show that 2(x+y+z)/a+b+c=(b+c)x+(c+a)y+(a+b)z/bc+ca+ab

Grade:12

1 Answers

Pratik Tibrewal
askIITians Faculty 37 Points
7 years ago
there is a property of ratio and proportion:
if a/x = b/y = c/z
then
a/x = b/y = c/z = (a+b+c)/(x+y+z)
similarly above eqn : (y+z)/(pb+qc) = (z+x)/(pc+qa) = (x+y)/(pa+qb) =

[ (y+z) + (z+x) + (x+y) ] / [ (pb+qc) + (pc+qa) + (pa+qb) ] = 2(x+y+z)/[(p+q) (a+b+c)] ----- (1)
multiply and divide first ratio by a, second ration by b, third ratio by c, and follow the same step
a(y+z)/a(pb+qc) = b(z+x)/b(pc+qa) = c(x+y)/c(pa+qb) =
[ a(y+z) + b(z+x) + c(x+y) ] / [ a(pb+qc) + b(pc+qa) + c(pa+qb) ]
= [(b+c)x + (c+a)y + (a+b)z] / [(bc + ca + ab)(p + q)] -------(2)
Equate eqn (1) and eqn (2) we get the desired result

Thanks and Regards,
Pratik Tibrewal
askiitans faculty
BTech IITG

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