Find the values of `a` for which one root of equation `(a-5)x^(2)-2ax+a-4=0` is smaller than 1 and the other greater than 2.

Explanation

The given equation can be written as
`x^(2)-((2a)/(a-5))x+((a-4)/(a-5))=0,a!=5`
Now let `f(x)=x^(2)-((2a)/(a-5))x+((a-4)/(a-5))`
As 1 and 2 lie between the roots of `f(x)=0`, we can take `Dgt0,1.f(1)lt0` and `1.f(2)lt0`.
(i) Consider `Dgt0`
`(-((2a)/(a-5)))^(2)-4.1.((a-4)/(a-5))gt0`
`implies(36(a-20/9))/((a-5)^(2))gt0 [ :' a1=5]`
or `agt20/9`.......i
(ii) Consider `1.f(1)lt0`
`1^(2)-((2a)/(a-5))+((a-4)/(a-5))lt0implies9/((a-5))gt0` or `agt5`..........ii
(iii) Consider `1.f(2)lt0`
`4-(4a)/((a-5))+((a-4)/(a-5))lt0`
`implies((4a-20-4a+a-4))/((a-5))lt0implies((a-24))/((a-5))lt0`
Or `5 ltalt24`......iii
Hence the values of `a` satisfying Eqs (i),(ii) and (iii) at the same time are `a epsilon (5,24)`.