Question

If x/a =y/b=z/c prove the follong photo​

Answer 1

See Explanation

Explanation:

Given

`(x)/(a)= (y)/(b) =(z)/(c)`

`Let \: (x)/(a)= (y)/(b) =(z)/(c)=m`

(where m is any constant)

`\implies`

`(x)/(a)=m\implies x =am`

`(y)/(b)=m\implies y =bm`

`(z)/(c)=m\implies z =cm`

To Prove:

`(ax-by)/((a+b)(x-y))+(by-cz)/((b+c)(y-z))=2`

LHS
`=(ax-by)/((a+b)(x-y))+(by-cz)/((b+c)(y-z))`

(Plug the values of x, y and z in the form of am, bm and cm respectively in the LHS)

`=(a(am)-b(bm))/((a+b)(am-bm))+(b(bm)-c(cm))/((b+c)(bm-cm))`

`=(a^2m-b^2m)/((a+b)m(a-b))+(b^2m-c^2m)/((b+c)m(b-c))`

`=(m(a^2-b^2))/(m(a+b)(a-b))+(m(b^2-c^2))/(m(b+c)(b-c))`

`=\frac{\cancel{m(a^2-b^2)}}{\cancel{m(a^2-b^2)}}+\frac{\cancel{m(b^2-c^2)}}{\cancel{m(b^2-c^2)}}`

= 1 + 1

= 2

= RHS

`\implies \purple{\bold{(ax-by)/((a+b)(x-y))+(by-cz)/((b+c)(y-z))}}=\red{\bold{2}}`

Thus proved