If the domain of log x-1 (2x^2-9x+4/x^2-4x+5) is (α, ∞) and log5(18x-x^2-77) is (β, γ), then the value of α^2 + β^2 + γ^2 is

Question

If the domain of \(\log _{x-1}\left(\frac{2 x^2-9 x+4}{x^2-4 x+5}\right)\) is \((\alpha, \infty)\) and \(\log_ 5\left(18 x-x^2-77\right)\) is \((\beta, \gamma),\) then the value of \(\alpha^2+\beta^2+\gamma^2\) is

Answer 1

Answer is "186"  

\(\frac{2 x^2-9 x+4}{x^2-4 x+5}>0\)       ....(i)

\(x-1>0, x-1 \neq 1 \)

\( \Rightarrow (2 x-1)(x-4)>0\) 

\(\therefore x \in(4, \infty) \)

\(\therefore \alpha=4 \)

\(\log 5\left(18 x-x^2-77\right) \)

\(\Rightarrow 18 x-x^2-77>0 \)

\(\Rightarrow x^2-18 x+77<0 \)

\(\Rightarrow(x-7)(x-11)<0 \)

\(x \in(7,11) \)

\( \therefore \beta=7, \gamma=11 \)

\(\therefore \alpha^2+\beta^2+\gamma^2 \)

= 16 + 49 + 121 

= 186