Question

If x/y=a+b/a-b then prove that x(x+y)/y(x-y)=a(a+b)/b(a-b)

(PLEASE SOLVE THE PROBLEM WITH CORRECT ANSWER)​

Answer 1

$\underline{\underline{\sf{\maltese\:\:Question}}}$

$\sf{\bullet\:\:If\:\: \dfrac{x}{y}=\dfrac{a+b}{a-b}}\textsf{ then prove that } \sf{\dfrac{x(x+y)}{y(x-y)}=\dfrac{a(a+b)}{b(a-b)}}$

$\underline{\underline{\sf{\maltese\:\:Given}}}$

$\sf{\bullet\:\:\dfrac{x}{y}=\dfrac{a+b}{a-b}}$

$\underline{\underline{\sf{\maltese\:\:To \:\:Prove}}}$

$\sf{\bullet\:\:\dfrac{x(x+y)}{y(x-y)}=\dfrac{a(a+b)}{b(a-b)}}$

$\underline{\underline{\sf{\maltese\:\:Answer}}}$

$\sf{\dfrac{x}{y}=\dfrac{a+b}{a-b}}\:\:\:\:\bf{........(1)}$

$\textbf{Using Componendo and Dividendo}\:\bf{:If\:\dfrac{a}{b}=\dfrac{c}{d},\:then\:\dfrac{a + b}{a - b}=\dfrac{c + d}{c - d} }$

$\implies\sf{\displaystyle \frac{x+y}{x-y}=\frac{a+b+a-b}{a+b-a+b}}$

$\implies\sf{\displaystyle\frac{x+y}{x-y}=\frac{2 a}{2 b}}$

$\implies\sf{\dfrac{x+y}{x-y}=\dfrac{9}{6}}\:\:\:\:\bf{........(2)}$

$\textbf{Multiplying (1) and (2)}$

$\implies\sf{\displaystyle\frac{x}{y} \times\frac{x+y}{x-y}=\frac{a+b}{a-b} \times \frac{a}{b}}$

$\implies\sf{\displaystyle\frac{x\left(x+y\right)}{y\left(x-y\right)}=\frac{a\left(a+b\right)}{b\left(a-b\right)}}$

$\textbf{Hence Proved !!!}$