Question

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

Answer 1

Answer:

See Explanation

Step-by-step explanation:

Given

$\frac{x}{a}= \frac{y}{b} =\frac{z}{c}$

$Let \: \frac{x}{a}= \frac{y}{b} =\frac{z}{c}=m$

(where m is any constant)

$\implies$

$\frac{x}{a}=m\implies x =am$

$\frac{y}{b}=m\implies y =bm$

$\frac{z}{c}=m\implies z =cm$

To Prove:

$\frac{ax-by}{(a+b)(x-y)}+\frac{by-cz}{(b+c)(y-z)}=2$

LHS $=\frac{ax-by}{(a+b)(x-y)}+\frac{by-cz}{(b+c)(y-z)}$

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

$=\frac{a(am)-b(bm)}{(a+b)(am-bm)}+\frac{b(bm)-c(cm)}{(b+c)(bm-cm)}$

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

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

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

= 1 + 1

= 2

= RHS

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

Thus proved