Let (a, b) ⊂(0,2 π) be the largest interval for which sin^-1( sin θ)- cos^-1( sin θ)>0, θ ∈(0,2 π), holds. If α x^2+β x+ sin^-1(x^2-6 x+10)+…

Let $\displaystyle (a, b) \subset(0,2 \pi)$ be the largest interval for which $\displaystyle \sin ^{-1}(\sin \theta)-\cos ^{-1}(\sin \theta)>0, \theta \in(0,2 \pi)$, holds. If $\displaystyle \alpha x^2+\beta x+\sin ^{-1}\left(x^2-6 x+10\right)+\cos ^{-1}\left(x^2-6 x+10\right)=0$ and $\displaystyle \alpha-\beta=\mathrm{b}-\mathrm{a}$, then $\displaystyle \alpha$ is equal to: