Question

If the equation $\frac{ ax ^2-24 x + b }{ x ^2-1}= x$ has exactly two real solutions and their sum is 12 then find the value of $(a-b)$.

Answer 1

$ 2 \alpha+\beta= a $ .....(1)
$ \alpha^2+2 \alpha \beta=23 $ .....(2)
$\text { and } \alpha^2 \beta=\beta $ .....(3)
$\text { Also given } \alpha+\beta=12 $ .....(4)
$\text { from (2) and (4) }$
$\alpha^2+2 \alpha(12-\alpha)=23 $
$\alpha^2+24 \alpha-2 \alpha^2=23 $
$\alpha^2-24 \alpha+23=0 $
$\alpha=1 \text { (rejected) } $
$ \text { since } x \neq \pm 1$
$\therefore \alpha=23 ; $
$ \therefore \beta=-11 $
$\therefore a = 3 5 \text { from }(4)$
$\text { and } b=\alpha^2 \beta=529 \times-11 $
$\Rightarrow b=- 5 8 1 9 $
$\Rightarrow a - b =35-(-5819)=5854$