Question

Solve :$\frac{a}{ax-1}+\frac{b}{bx-1}-a-b=0$.If $x=\frac{m}{a+b}$.find $m$

Answer 1

Solution:-

Given:-

$x=\frac{m}{a+b}$

$m=?$

$\frac{a}{ax-1}+\frac{b}{bx-1}-a-b=0$

$\Rightarrow \frac{a}{ax-1}-b=a-\frac{b}{bx-1}$

$\Rightarrow \frac{a-b(ax-1)}{ax-1}=\frac{a(bx-1)-b}{bx-1}$

$\Rightarrow \frac{a-abx+b}{ax-1}=\frac{abx-a-b}{bx-1}$

$\Rightarrow (a-abx+b)(bx-1)=(abx-a-b)(ax-1)$

$\Rightarrow (bx-1)(a+b-abx)=-(ax-1)(a+b-abx)$

$\Rightarrow (bx-1)(a+b-abx)+(ax-1)(a+b-abx)=0$

$\Rightarrow (a+b-abx)(ax-1+bx-1)=0$

$\Rightarrow (a+b-abx)((a+b)x-2)=0$

$\Rightarrow a+b-abx=0\text{or}(a+b)x-2=0$

$\Rightarrow x=\frac{a+b}{ab}\text{or}x=\frac{2}{a+b}$

As given that,

$x=\frac{m}{a+b}$

$\Rightarrow$ On comparing the above valuee of x with $x=\frac{2}{a+b}$, we have

$m=2$

Hence, the correct answer is 2.