Question

If $\dfrac{{3x}}{{(x - a)(x - b)}} = \dfrac{2}{{(x - a)}} + \dfrac{1}{{(x - b)}}$, then $a:b = $
A. $1:2$
B. $ - 2:1$
C. $1:3$
D. $3:1$

Answer 1

Hint:Let LHS be $\dfrac{{3x}}{{(x - a)(x - b)}}$ and RHS be $\dfrac{2}{{(x - a)}} + \dfrac{1}{{(x - b)}}$.Now solve RHS by taking LCM and comparing the numerator, you will get the relation between $a,b$.

Complete step-by-step answer:
Here we are given that
$\dfrac{{3x}}{{(x - a)(x - b)}} = \dfrac{2}{{(x - a)}} + \dfrac{1}{{(x - b)}}$
Now here we have on
LHS is $\dfrac{{3x}}{{(x - a)(x - b)}}$
RHS is $\dfrac{2}{{(x - a)}} + \dfrac{1}{{(x - b)}}$
Upon solving RHS and taking the LCM, we get that
$\dfrac{{2(x - b) + 1(x - a)}}{{(x - a)(x - b)}}$
Now solving we get that
$\dfrac{{2x - 2b + x - a}}{{(x - a)(x - b)}}$
$\dfrac{{3x - 2b - a}}{{(x - a)(x - b)}}$
So we know that
LHS=RHS
$\dfrac{{3x}}{{(x - a)(x - b)}}$$ = $$\dfrac{{3x - 2b - a}}{{(x - a)(x - b)}}$
We get on comparing that
$3x = 3x - 2b - a$
$a = - 2b$
$\dfrac{a}{b} = \dfrac{{ - 2}}{1}$
So we get that $a:b = - 2:1$

Note:The main key step for solving this question is simplifying the L.H.S and R.H.S by taking LCM and solving the numerator part.Students may multiply the denominator part which gives quadratic equation and try to simplify it but it is time consuming and gets complicated, also may calculation get wrong .So always try to reduce the expression and simplify it.